Modelingcolloidalnanoparticles: Fromgrowthtodeposition · Modelingcolloidalnanoparticles: Fromgrowthtodeposition Joakim Löfgren isbn 978-91-7905-176-1 ©JoakimLöfgren,2019 DoktorsavhandlingarvidChalmerstekniskahögskola

thesis for the degree of doctor of philosophy

Modeling colloidal nanoparticles:From growth to deposition

Joakim Löfgren

Department of Physicschalmers university of technology

Göteborg, Sweden 2019

Modeling colloidal nanoparticles:From growth to depositionJoakim Löfgren

isbn 978-91-7905-176-1

© Joakim Löfgren, 2019

Doktorsavhandlingar vid Chalmers tekniska högskolaNy serie nr 4643ISSN 0346-718X

Department of PhysicsChalmers University of Technologyse-412 96 Göteborg, SwedenTelephone +46317723252

Cover: The Blasphemous Monstrosity, a small, thiolated gold nanoparticle [1].Chalmers reproserviceGöteborg, Sweden 2019

Modeling colloidal nanoparticles:From growth to deposition

Joakim LöfgrenDepartment of Physics

Chalmers University of Technology

Abstract

In recent decades metal nanoparticles (NPs) have been the subject of intense research.The interest stems from the NPs physicochemical properties that can be convenientlytuned through, e.g., their size, shape or composition. A good example is the selectiveabsorption of electromagnetic radiation exhibited by gold nanorods, which is lever-aged for applications in sensing and medicine. In order to realize the full potential oftechnologies reliant on NPs and ensure fitness for commercial use, facile fabricationmethods that allow for a high degree of shape and size control are required. For thispurpose, wet-chemistry-based synthesis in which colloidal NPs self-assemble into atargeted morphology have emerged as promising candidates. Development and refine-ment of synthesis protocols is, however, hampered by a lack of theoretical understand-ing of the complex chemical environment in NP solutions. As a result, experimentalworkers are often left to rely on intuition. This applies not only to the growth processitself, but also later down the processing chain, e.g., during NP deposition.

This thesis aims to address two problem areas relating to NP growth and depositionwhere current models need improvement. The first such area concerns the descrip-tion of ionic and molecular adsorption on the surface of metal NPs. We show howcombining thermodynamic modeling, density functional theory and experimental datacan lead to more realistic NP shape predictions. A closely related subject is the growthmechanism of anisotropic gold nanorods, which has been a subject of debate for almostthe three decades. Here, we consider possible avenues through which shape anisotropycan arise using insight frommolecular dynamics simulations. The second problem areais the description of forces between NPs and nearby surfaces, which is relevant, e.g, forapplications reliant on NP deposition. A model based on Derjaguin-Landau-Verwey-Overbeek theory is developed that describes how the shape and composition of a sur-face affects particle deposition.

Keywords: colloidal nanoparticles, gold, anisotropic, nanorods, modeling, density func-tional theory, thermodynamics, dispersive interactions, van derWaals, deposition, DLVO

list of appended papers

This thesis is based on work presented in the following papers:

I Understanding the Phase Diagram of Self-Assembled Monolayers of Alka-nethiolates on GoldJoakim Löfgren, Henrik Grönbeck, Kasper Moth-Poulsen, and Paul ErhartThe Journal of Physical Chemistry. C, 120, 12059 (2016)

II Quantitative assessment of the efficacy of halides as shape-directing agentsin nanoparticle growthJoakim Löfgren, Magnus Rahm, Joakim Brorsson, and Paul ErhartIn preparation

III Modeling the growth of gold nanorodsNarjes Khosravian, Joakim Löfgren and Paul ErhartIn preparation

IV Understanding Interactions Driving the Directed Assembly of Colloidal Nanopar-ticlesJohnas Eklöf-Österberg, Joakim Löfgren, Paul Erhart, Kasper Moth-PoulsenSubmitted for publication

V libvdwxc: a library for exchange�correlation functionals in the vdW-DF familyAsk Hjorth Larsen, Mikael Kuisma, Joakim Löfgren, Yann Pouillon, Paul Erhart, andPer HyldgaardModelling and Simulation in Materials Science and Engineering, 25, 065004 (2017)

Publications not included in this thesis

The following publications are outside the scope of this thesis:

Electric field-controlled reversible order-disorder switching of a metal surfaceLudvig de Knoop, Mikael Juhani Kuisma, Joakim Löfgren, Kristof Lodewijks, MattiasThuvander, Paul Erhart, Alexander Dmitriev and Eva OlssonPhysical Review Materials 2, 085006 (2018)

The author’s contribution to the papers:

I The author carried out the calculations, analysis and wrote most of the paper.

II The author carried out the calculations, analysis and wrote most the paper.

III The author wrote the code, carried out the calculations, theoretical analysis and wrotepart of the paper.

IV The author participated in the analysis of the calculations and wrote part of the paper.

V The author contributed to the benchmarking calculations, testing of the software andwriting of the paper.

vi

Contents

List of Abbreviations ix

1 Introduction 11.1 Metal nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Challenges for nanotechnology . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis aim and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Colloidal nanoparticles 52.1 Historical highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Fabrication methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Seeded growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Gold nanorods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Elements of nanoparticle modeling 153.1 Equilibrium shapes of nanoparticles . . . . . . . . . . . . . . . . . . . . 153.2 A thermodynamic model of adsorption . . . . . . . . . . . . . . . . . . 16

3.2.1 Realization using DFT . . . . . . . . . . . . . . . . . . . . . . . 183.3 Interparticle forces and DLVO theory . . . . . . . . . . . . . . . . . . . 23

3.3.1 van der Waals interactions . . . . . . . . . . . . . . . . . . . . . 243.3.2 Electrostatic interactions . . . . . . . . . . . . . . . . . . . . . . 253.3.3 Surface element integration . . . . . . . . . . . . . . . . . . . . 27

4 First-principles calculations 294.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . 314.1.2 Kohn-Sham theory . . . . . . . . . . . . . . . . . . . . . . . . . 324.1.3 Approximating the exchange-correlation functional . . . . . . 334.1.4 Dispersion-corrections . . . . . . . . . . . . . . . . . . . . . . . 344.1.5 Solving the Kohn-Sham equations . . . . . . . . . . . . . . . . 36

vii

4.1.6 Periodic systems and plane waves . . . . . . . . . . . . . . . . . 37

5 Summary of appended papers 395.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A Origin of the dispersive interaction 47

Acknowledgments 51

Bibliography 53

Papers I-V 61

List of Abbreviations

AR aspect ratio 9, 14

CMC critical micelle concentration 9

CTAB cetyl trimethyl ammonium bromide 9–11

DFT density functional theory vii, 18–20

DLVO Derjaguin-Landau-Verwey-Overbeek 23, 24, 27

EDL electrostatic double layer 23, 27

LPBE linearized Poisson-Boltzmann equation 26

LSPR localized surface plasmon resonance 11, 12

NP nanoparticle 1–7, 11–13, 15, 16, 18, 23, 28

PB Poisson-Boltzmann 26, 27

SEI surface element integration 27, 28

vdW van-der-Waals 23, 24, 27, 28

ZPE zero-point energy 19

ix

1Introduction

Nanoscience is defined as the study and development of systems having one or moredimensions lying in the range of 1 to 100 nm. At this length scale quantum mechani-cal effects can lead to new or altered properties of a material that do not manifest onmacroscopic scales.

Historically, the use of nanoscale systems long predates any theory or systematicattempt at understanding the underlying phenomena, as evidenced, e.g., in the stainedglass of church windows, resulting from the interaction of light with nanoparticlespresent in the glass [2]. Only with the advent of modern physics during the 20th cen-tury has theoretical understanding, manufacturing and precise manipulation of suchsystems become possible. Today, nanoscience constitutes an enormous research fieldwith applications ranging from consumer grade products to next generation technolo-gies in electronics [3–6], medicine [7–9] and renewable energy systems [10–12].

1.1 Metal nanoparticlesMetal nanoparticles (NPs) represent a prominent family of nanosystems, consisting ofsolid particles of various shapes where all dimensions are in the nanometer range. Inapplications, NPs can assume a multitude of roles, e.g., functioning as building blocks,delivery agents, catalysts, sensing devices or therapeutic agents. Below, we elaborateon these uses and discuss how they relate to different NP properties. Extra emphasis isput on the effect of physical properties such as shape, size, composition, that can con-veniently be tuned during fabrication. Indeed, the versatility offered by such “handles”is one of the primary driving forces behind the extensive research into NPs.

For the use of NPs as building blocks or delivery agents, the reactivity of its surface,or more precisely the propensity of atoms and molecules to adsorb on the surface is one

1

Chapter 1. Introduction

a) b) c)

Figure 1.1: Various nanomaterials and their applications. a) Hydrogen sorption in Pdenables the development of hydrogen sensing devices. b) The tunable optical propertiesof Au nanorods can be used for cancer treatment. c) Carbon nanotubes have excellentmechanical properties that can enhance the strength of materials.

of the most important considerations. Adsorbed molecules can act as linkers betweennanoparticles to build circuits [13, 14] or as carriers that pharmaceuticals can “piggy-back” onto and create drug delivery systems [15–17]. While the origin of metal surfacereactivity can be surprisingly subtle in many cases [18], computational methods cannowadays aid us in understanding and predicting its dependence on parameters suchas NP shape and composition.

An oft-mentioned fact about NPs is the high surface-to-volume ratio found in a sam-ple of NPs compared to an otherwise equivalent bulk sample. The usefulness of havinga large amount of surface area is easy to appreciate if we consider heterogenous cataly-sis, where the use of NPs has been actively studied since the middle of the 20th century[19]. Here, transient binding of reactants to catalytically active sites on the NP surfacegives rise to new reaction pathways that can lower the reaction energy barrier. NP-based catalysts thus have the potential to greatly expedite reactions due to the largeamounts of surface area exposed.

Another large group of applications are enabled by the scattering, adsorption or emit-ting of electromagnetic radiation by NPs. Within the context of this thesis, a highlyrelevant example is provided by gold nanorods (Fig. 1.1b). Like silver [20], gold NPs ex-hibit strong localized surface plasmon resonances (LSPRs) that allow them to efficientlyabsorb incoming light in certain frequency windows [21]. In the case of nanorods, theshape anisotropy leads to a splitting of the LSPR into one longitudinal and one transver-sal mode. The frequency of the longitudinal resonance mode is tunable through the

2

1.2. Challenges for nanotechnology

aspect ratio of the rod, providing control over the optical response of the rod. This isthe basis of photothermal cancer therapy where nanorods are targeted at a tumor andsubsequently irradiated at the LSPR frequency. The adsorbed light is then given off asheat to the environment, resulting in the destruction of the cancer cells [22, 23]. In asimilar vein, Pd NP have been investigated as a means for building hydrogen sensors.The construction of such sensors is crucial for the use of hydrogen as a renewable en-ergy source since fuel leakage can easily lead to volatile mixtures of hydrogen and air.The key idea is that hydrogen can be stored in the interstitial sites of Pd (Fig. 1.1b).The amount of hydrogen stored depends on the partial pressure of hydrogen in thesurrounding gas-phase, and any changes resulting from a sudden release of hydrogencan be detected as shift in the peak of the NPs LSPR spectrum [24].

1.2 Challenges for nanotechnologyDespite the potential of NP-based technologies, many of the applications are still intheir infancy. There is thus no shortage of challenges that have to be addressed beforethey can be adopted for widespread use. One challenge concerns up-scaling; industrialproduction requires cost-effective manufacturing processes while historically NPs havebeen created in specialized laboratories using time-consuming methods and expensivematerials. In the recent decades, top-downmethods have emerged as a scalable route toobtaining nanomaterials for use in, e.g., electronics. They are, however, less suited forNPs since they often rely on surface modification via lithography. On the other hand,bottom-up, wet chemical synthesis offers a simple way of obtaining NPs where prop-erties of the products such as size, shape and composition are tunable through varioussynthesis parameters and components. A key issue here, however, is the lack of the-oretical understanding concerning the synthesis processes themselves. Consequentlyimprovements to protocols are mainly due to heuristics and ad hoc experimentationrather than the result of a rational design process, an approach that is ultimately lim-ited.

From a societal perspective, there are also safety issues regarding nanotechnologyas a whole, including toxicity, environmental pollution and the development of nano-material-basedweaponry. For instance, the interactions of NPs with complex biologicalenvironments such as the inside of a human body are in general not well understood.The small size of the particles facilitates transport through the body and provides alarge available surface for interactions with surrounding biomolecules. Indeed, studieshave concluded that NPs entering the human body through the airways [25] or gastro-intestinal tract [26] can be taken up in the blood circulation and distributed to variousorgans via the liver and the spleen. The clearance rate of NPs from the body depends onthe specific particle properties and may proceed via different mechanisms [27]. Longterm retention of NPs leads to concerns regarding, e.g., cell damage as the result of

3

Chapter 1. Introduction

excess generation of reactive oxygen species due to the presence of the particles [28].There is thus a need for detailed experimental as well as theoretical studies with regardsto the interactions in the interfacial region between NP surface and organic molecules.

1.3 Thesis aim and outlineThe overarching goal of this thesis is to help bridge the gap between theoretical mod-eling and application for a number of topics related to the challenges described inSect. 1.2. Most prominently featured are issues related to adsorption of molecules andions at metal surfaces, where the importance of properly accounting for dispersive in-teractions and solvation effects is investigated in Papers I and II. From a more technicalperspective, the incorporation of dispersive interactions into density functional theory(DFT), which is extensively used throughout the first two papers, is described in PaperV. While the ability to model NP surfaces from first-principles methods such as DFT isextremely useful in modeling the growth of NPs, the length and time scales involvedoften necessitate the use of other methods, e.g., molecular dynamics (MD) in order togain a more complete understanding. This is the case for gold nanorod growth, wherethe origin of their anisotropy and the growth mechanism has been under debate for al-most two decades. Here, Paper III presents a critical review of recent modeling efforts,primarily based on MD. Finally, in Paper IV, another step upwards in the length andtime scales is taken as the modeling of NP deposition is addressed using Derjaguin-Landau-Verwey-Overbeek (DLVO) theory.

A background on colloidal NPs including a brief historical perspective, fabricationmethods, stability, and optical properties is provided in Chapter 2. In particular, a gen-eral overview of wet-chemical synthesis of NPs via seed-mediated growth is providedand subsequently exemplified by the synthesis of gold nanorods. Chapter 3 begins withan overview of various aspects of colloidal modeling and relevant methodology. Thenext two sections provide background for the two principal modeling aspects most rel-evant for the thesis. The first such aspect is the thermodynamics of surfaces and NPshape prediction, where a detailed account is also provided for how to combine the de-rived expression with DFT. The second aspect concerns interparticle and surface forces,focusing on DLVO theory. Since DFT calculations play a central role in this thesis, theunderlying theory is presented separately in Chapter 4. Here, an important topic isthat of dispersion-corrections, which play a prominent role in several of the appendedpapers, which are summarized in Chapter 5.

4

2Colloidal nanoparticles

Elements combine and change into compounds. But that’s all oflife, right? It’s the constant, it’s the cycle. It’s solution,dissolution. Just over and over and over. It is growth, then decay,then transformation.

Walter White

2.1 Historical highlightsThe scientific study of colloidal NPs began in the 1850s with Michael Faraday, who wasinterested in the optical properties of thin gold sheets [29]. As a by-product of thisresearch he obtained a gold sol and began a systematic investigation of colloidal goldcreated from the reduction of a solution of gold chloride by phosphorus. He deduced,among other things, that the characteristic red tint of the sol was due to the interactionof light with suspended gold particles invisible to the naked eye [29]. Colloid science asa whole, however, did not start to pick up momentum until the end of the 19th century.In particular regarding NPs there are several highlights from this period. The firstsynthesis of colloidal silver from reduction of silver citrate by iron(II) citrate was carriedout in 1889 by Lea [30]. Building on the work of Faraday, Zsigmondy invented thenucleus method in 1905 for the synthesis of gold sols [31]. Chloroauric acid (HAuCl4)was reduced by white phosphorus creating a solution of NPs seeds 1–3 nm in diameter.To obtain larger NPs a growth solution was prepared in a vial and initially kept separatefrom the seeds. This growth solution containedmore gold salt, which was reduced withformaldehyde. Addition of the seeds to this growth solution resulted in the formationof gold NPs 8 to 9 nm in diameter. Conceptually, Zsigmondy’s method is a predecessor

5

Chapter 2. Colloidal nanoparticles

to much of modern colloidal NP synthesis that follow the same logic of letting seedparticles grow into a target morphology.

Significant advancements were also made on the theoretical side. The rapid motionundergone by suspended colloidal particles and their stability against gravity was ex-plained in terms of Brownian motion by Einstein [32] and Smoluchowski [33]. Theoptical properties of metal sols were elucidated by Mie [34] and Gans [35], who de-rived shape-dependent expressions for the scattering and absorption of incident lightby the particles from Maxwell’s equations, providing an explanation for the observa-tions made by Faraday some fifty years earlier.

2.2 Fabrication methodsFabrication of nanomaterials is often categorized as being either top-down or bottom-up. A top-down method starts from a bulk sample and then successively removes ma-terial until the desired shape or form is obtained as done in, for instance, optical orelectron lithography. Conversely, bottom-up fabrication is the assembly of a nanoma-terial from smaller constituent pieces. Both categories of nanofabrication have theirdistinct advantages and limitations, but only bottom-up, wet-chemistry approacheswill be dealt with here since they provide the most facile route to obtaining NPs whiletop-down methods are more suitable for, e.g., surface patterning [36].

There are two principal approaches to bottom-up fabrication: gas-phase and wet-chemical (liquid-phase) synthesis. A gas-phase method starts from a precursor whichis evaporated, forming an intermediate state containing monomers¹. Nucleation in thisintermediate state then leads to primary particles that form NPs, e.g., by coalescing[37]. The resulting colloid of solid particles suspended in a gas is known as an aerosol.Liquid-phase synthesis processes follow a similar pattern. A precursor is created, forinstance by dissolution of a solid, and an intermediate state of monomers is producedfrom chemical reactions. Nucleation in the monomer solution then leads to primaryparticles that subsequently form NPs through e.g., diffusive growth [38]. Generally,gas-phase routes yield very pure product particles with minimal by-products, and scalebetter than their liquid-phase relatives. When it comes to the fabrication of anisotropicshapes, however, wet-chemical synthesis is the superior approach.

One example of a wet-chemical synthesis protocol has already been encountered,namely Zsigmondy’s nuclear method described in the previous section. Another rele-vant example is the Turkevich method [39], which is one of the most commonly em-ployed protocols for the synthesis of spherical gold NPs². A solution of chloroauricacid is heated to its boiling point and then a solution of sodium citrate (Na3C6H5O7) isadded. The sodium citrate will act both to reduce the gold and as a capping agent for

¹In this text, a monomer refers to a unit of the precursor compound that supplies the growth.²It can also be used to obtain silver NPs.

6

2.2. Fabrication methods

the subsequently formed NPs, electrostatically stabilizing them against agglomerationby forming a negatively charged surface layer.

2.2.1 Seeded growth

Shape-controlled wet-chemical synthesis of metal NPs is perhaps most easily achievedbased on the concept of seeded, or seed-mediated, growth. Much like in Zsigmondy’snucleus method, the key characteristic of seeded growth is the preparation of a NPseed solution, which is later added to a separately prepared growth solution in orderto achieve the desired product. The decoupling of the nucleation and growth stagesallows for better control over size and shape of the final particles, at the cost of a morelaborious synthesis process compared to one-pot protocols. Modern seeded growth be-gan with the synthesis of gold nanorods in 2001 by Murphy and coworkers [40], whoalso demonstrated that it could be used to achieve greater size control when synthe-sizing spherical gold NPs [41]. Following the same general principles, seed-mediatedgrowth protocols have since been reported for a variety of other metals such as Ag, Pd,Pt, and Cu [42–44]. In a typical seeded growth protocol the seed solution will consist ofa metal salt, a reducing agent and a capping (stabilizing) agent (Fig. 2.1) to prevent ag-gregation of the seeds and possibly alter the surface energetics to expose certain facets.The growth solution has the same basic ingredients but different compounds may beused depending on the goal of the synthesis. For instance, another type of metal saltmay be used if bimetallic structures are targeted [45]. Furthermore, the growth solu-tion reducing agent must be sufficiently weak to prevent nucleation of additional seeds.This allows for metal ions to be reduced in a catalytic reaction near the surface of theseeds [40]. The resulting growth of the pre-existing seeds can then be guided towardsanisotropic shapes by adding a second capping agent. In practice, the basic synthesisrecipe described here is often augmented with additives or co-surfactants in order toimprove yield, monodispersity or to achieve different shapes. In terms of the growthstage, one can distinguish between one-step and multi-step protocols [46]. The latterare characterized by several iterated growth cycles, during whichmore growth solutionis added to aliquots drawn from the current seed/growth solution mix.

Deciphering the precise role played by the various ingredients is very difficult asthey may vary on a method-to-method basis and synergistic effects can make the in-dividual impact of a compound hard to disentangle from the whole. The descriptionsprovided above, while intuitive, are too simplistic to adequately account for the myriadpermutations of seeded growth protocols.

7


Metal salt 1Cappring agent 1

Reducing agent 1

Nanoparticle seeds

Metal salt 2Cappring agent 2

Reducing agent 2

Monomers Nanoparticles

Seed solution

Growth solution Product

Add seeds

Figure 2.1: A general outline for seed-mediated growth protocols. Decoupling the nu-cleation and growth stages by separately synthesizing seeds and then adding them to agrowth solution of monomers allows for more precise shape and size control. Depend-ing on the desired outcome, metal salt and capping agents can vary between the twosolutions. The growth solution reducing agent is constrained by the requirement thatno additional nucleation take place.

2.3 Gold nanorods

Gold nanorods are the most extensively studied synthesis products of seed-mediatedgrowth owing to their physicochemical properties, spurring research into a multitudeof potential applications in, e.g., drug delivery, cancer therapy and sensing. Further-more, they also pose a very interesting problem from a theoretical perspective withmany questions that remain unanswered regarding, e.g., the origin of the symmetrybreaking event that occurs in the evolution from spherical seeds to nanorods. As thefirst seeded growth method to gain widespread popularity it is also particularly appro-priate as a concrete illustration of the generalized description presented in Sect. 2.2.1.

8

2.3. Gold nanorods

2.3.1 SynthesisTurning to the actual nanorod synthesis procedure, it should be noted that since itsadvent many variations of the original protocol have spawned that try to improve,e.g., yield or control over the AR. The latter is especially important since it allows fortuning the plasmonic response (Sect. 2.4). Rather than providing an exhaustive list ofall possibilities, focus will be on the original three-step protocol by Murphy [40] andthe nowadays more commonly employed silver-assisted one-step protocol [47].

Three-step protocol (P-1). This protocol follows the general seeded growth pro-cedure outlined in Fig. 2.1. The seeds are prepared by reducing HAuCl4with ice-coldsodiumborohydride³ (NaBH4) and adding (trisodium) citrate to stabilize the seeds againstaggregation. In the growth solution the reduction of HAuCl4is instead accomplishedwith ascorbic acid (AA) in the presence of CTAB surfactant (Fig. 2.2). A concentrationof around 0.1M CTAB is required in order to obtain nanorods, which is 100 times morethan the first CMC, 1.0mM, of CTAB at room temperature [48]. The rods are growniteratively in three steps and the final solution must be left for several hours to attainmaximal growth. The rods thus synthesized are oriented along [110] and have a penta-twinned structure. In an idealized model the side facets are all {100} (Fig. 2.3a), but inpractice a mix between {100} and {110} is often found [49]. The rod termination has apyramidal structure and consists of {111} facets (Fig. 2.3). The AR ranges between 10 to25 and the length of a fully grown rod can be in the lower micron range [50] (Fig. 2.4).A significant shortcoming with the P-1 protocol is that the initial yield of nanorodsby shape is only about 5% before purification. It turns out there is a simple way ofobtaining nanorods in high yield by a few straightforward modifications to P-1.Silver-assisted protocol (P-2). In this synthesis protocol, CTAB replaces sodium cit-

rate as the capping agent in the seed solution and silver nitrate (AgNO3) is added insmall amounts to the growth solution (Fig. 2.2). Nanorods are subsequently grown ina single step, yielding rods with ARs that can be precisely varied between 2 to 5 bychanging the silver concentration. The overall dimensions of these rods are signifi-cantly smaller compared to the P-1 rods, with typical rod lengths under 100 nm [50].Furthermore, the nanorods have an octagonal cross-section and are single-crystallinerather than penta-twinned. While no broad consensus has been achieved with regardsto the faceting, several studies suggest the presence of high-index prism facets such as{520} [51], or possibly a mix between such facets and {100}/{110} [52]. The terminationis truncated pyramidal and exposes a mix of alternating {111} and {110} facets. Manyattempts at further improvement of the silver-assisted growth protocol have beenmadeby introducing various additives such as organic molecules and acids. A notable exam-ple is the introduction of a co-surfactant in the form of sodium oleate (NaOl), whichhas been observed to produce nanorods with a 99.5% yield and tunable AR [53]. Inter-

³Due to the quick reaction kinetics of NaBH4, a lower temperature is necessary to obtain a controlledreaction.

9


Au

Cl

Cl

ClCl

_

H+

Chloroauric acid

(HAuCl4)

O

O

O

O

O

O

OH

Na+

Na+

Na+

-

-

-

Tri-sodium citrate (citrate)

N

CH3

H3C

H3C

H3C

14

Br-

+

Hexadecyltrimethyl-ammoniumbromide (CTAB)

B

H

H

H

H

_

Na+

Sodium borohydride

(NaBH4)

O O

OHOH

H

OH

OH

Ascorbic acid (AA)

Silver nitrate

(AgNO3)

Seed solution Growth solution

P-1 P-2

Chloroauric acid

(HAuCl4)

Hexadecyltrimethyl-ammoniumbromide (CTAB)

N

O

O O

_

Ag+

or

P-2

Figure 2.2: Chemical structural formulas and names of the compounds used in theseeded growth of gold nanorods via either the P-1 or the P-2 protocol. In a P-2 synthe-sis, the CTAB replaces citrate as the stabilizing agent in the seed solution and a smallamount of silver nitrate is added to the growth solution.

a)

Penta-twinned

Pentagonal

{100

}

<110>

{111}{111}

{100

}

<110>

b)

Single-crystalline

<100>

{111} {110}{110}

{100}

Octagonal

{520

}

{520

}

<100>

Figure 2.3: Proposed models, including cross-sections views, for the faceting of goldnanorods. a) The three-step protocol yields penta-twinned nanorods with a pyrami-dal termination. b) For the silver-assisted protocol, a single-crystalline structure withan octagonal cross-section is obtained. Different models have been proposed for thefaceting, shown here is one based on high-index {520} prism facets.

10

2.4. Optical properties

estingly, the amount of CTAB required for this synthesis is around three times lowerthan what is commonly used for nanorod synthesis.

Figure 2.4: The products of a P-1 synthesis imaged at an angle using a scanning electronmicroscope [54]. Note that in addition to high-aspect ratio gold nanorods, a variety ofother shapes is obtained as well.

2.4 Optical propertiesMuch of the scientific interest in NPs stems from the fact that they can exhibit LSPR,giving rise to shape-dependent absorption and scattering. Given sufficient understand-ing of the correlation between particle shape and the synthesis parameters, the opticalproperties of the NPs can thus be controlled. As previously mentioned, the mathemat-ical theory of interaction between light and colloidal particles started with Mie in 1908[34], who derived analytical expressions describing the absorption and scattering ofsmall spherical particle using classical electromagnetic theory. Only a brief summaryof some especially pertinent results will be given here, for a more complete treatmentthe reader is referred to, e.g., [55]. The LSPR can be understood as a collective oscilla-tion of electrons close to the surface of a NP that is excited by an external electric field(Fig. 2.5). The electrons behave similarly to a harmonic oscillator under the influence ofa harmonic external force and there exists a certain resonance frequency, for which theamplitude attains a maximum⁴. Consequently, light propagating through a NP sol willbe absorbed and scattered by the particles. After traversing a length 𝐿, the intensity ofthe light will have decayed according to the Beer-Lambert law

𝐼 (𝐿) = 𝐼0 exp(−𝜌𝑁𝐶ext𝐿), (2.1)

⁴A more apt analogy would be that of a damped driven harmonic oscillator in order to take theplasmon decay into account.

11


e-

e-

Electromagnetic wave

k

E

Electron cloud

εm

ε

Dielectric medium

Figure 2.5: A schematic illustration of a localized surface plasmon resonance in a NP.An external electromagnetic wave with the right frequency can a excite a collectiveoscillation of the electron cloud near the surface of the particle.

where 𝐼0 is the initial intensity, 𝜌𝑁 the number density, and 𝐶ext the so-called extinctioncross section, which is defined as the sum of the absorption and scattering cross sections𝐶ext = 𝐶ads + 𝐶sca. In the case of an electromagnetic wave incident on a sphericalparticle of radius 𝑎 and surrounded by a dielectric medium, Maxwell’s equations canbe solved and cross sections can be respresented in terms of infinite series. These arevery laborious to compute, but in the Rayleigh-limit where the scattering particle isassumed to be smaller than the wavelength 𝜆 of the incident light simple formulas canbe found. Assuming the particles are immersed in a medium with dielectric constant𝜖𝑚 and the external field has wavenumber 𝑘 = 2𝜋/𝜆 they read

𝐶abs = 4𝜋𝑘𝑎3Im { 𝜖 − 𝜖𝑚𝜖 + 2𝜖𝑚

} (2.2)

and

𝐶sca =8𝜋3 𝑘3𝑎6 ( 𝜖 − 𝜖𝑚

𝜖 + 2𝜖𝑚)2. (2.3)

Maximum extinction is achieved if the denominator |𝜖 + 2𝜖𝑚| in Eqs. (2.2) and (2.3)is minimized. Assuming Im{𝜖(𝜔)} varies slowly around the minimum, the conditionbecomes

Re{𝜖(𝜔)} = −2𝜖𝑚. (2.4)

Frequencies for which this relation is satisfied are termed LSPR frequencies.Further insight into the behavior of these oscillatory modes can be gained by as-

suming a free electron gas model for the conduction electrons in the metal NP. Thedielectric function 𝜖(𝜔) is then given by

𝜖(𝜔) = 1 − 𝜔2𝑝𝜔2 + 𝑖𝛾𝜔 , (2.5)

12

2.4. Optical properties

where 𝜔𝑝 is the plasma frequency and 𝛾 a damping factor. Substitution of Eq. (2.5) intothe plasmon resonance condition Eq. (2.4) shows that maximum extinction will occurfor frequencies that satisfy

𝜔max = 𝜔𝑝√2𝜖𝑚 + 1. (2.6)

From this equation it is clear that the location of the plasmon resonance depends on thedielectric environment of the particle and it is this effect that forms the theoretical basisfor the use of plasmonic NPs in sensing applications. Mie’s work was later extended

Wavelength

Extin

ctio

n

Longitudinal mode

Transversal mode

AR incre

ase

Figure 2.6: A schematic illustration of the extinction spectrum of a spheroidal NP fortwo different aspect ratios. The anisotropy causes the plasmonic resonance of the par-ticle to split into a transversal and a longitudinal mode. As the aspect ratio increase thelongitudinal resonance peak is redshifted, while only a small blueshift is observed forthe transversal peak.

by Gans, who considered the more general case of ellipsoidal particles [35]. This resultis of particular interest since an ellipsoid with axes 𝑎 > 𝑏 = 𝑐, i.e. a prolate spheroid,can be used to approximate a nanorod for which no analytical formulas are otherwiseavailable. Note that the scattering will now depend on the orientation of the rod withrespect to the incident electromagnetic wave. To describe colloidal nanorods it is thusappropriate to assume an ensemble of randomly oriented prolate spheroids and thencalculate the average cross section yielding

⟨𝐶ext⟩ =2𝜋𝑉 𝜖3/2𝑚

3𝜆 ∑𝑖∈{𝑎,𝑏,𝑐}

(𝜖/𝑃𝑖)2(Re{𝜖} + ((1 − 𝑃𝑖)/𝑃𝑖)𝜖𝑚)2 + Im{𝜖}2

. (2.7)

13


Here, 𝑉 is the volume of a spheroid and 𝑃𝑖 are shape-dependent factors given in termsof the eccentricity 𝑒 = √(𝑎2 − 𝑏2)/𝑎2 with

𝑃𝑎 =1 − 𝑒2𝑒2 ( 1

2𝑒 log (1 + 𝑒1 − 𝑒 ) − 1) (2.8)

and

𝑃𝑏 = 𝑃𝑐 =1 − 𝑃𝑎

2 . (2.9)

Equation (2.7) yields extinction spectra that display two peaks. The single plasmon res-onance found for a spherical particle is thus split into two resonances, correspondingto one transverse and one longitudinal mode, respectively. It is interesting to studythe spectral shift of the resonances as a function of the aspect ratio AR = 𝑎/𝑏 of thespheroids (Fig. 2.6). While the transverse mode is largely unaffected except for a minus-cule blueshift, the longitudinal mode experiences a large redshift as the AR increases.This behavior is indeed experimentally observed for real gold nanorods, and the tun-ability of the longitudinal resonance peak is one of the main driving forces behind theextensive research surrounding gold nanorods as it holds the key to many applications.

14

3Elements of nanoparticle modeling

Due to the vast number of different time and length scales found among processesrelevant in NP modeling, there is an equally vast number of methods available for theirstudy. Rather than giving an overview of the field ¹ this chapter focuses on two specificelements of NP modeling relevant for the thesis. The first such element concerns theshape adopted by a NP in equilibrium and the related area of surface thermodynamics.In particular, this element forms the conceptual basis for Papers I and II and has majorimplications for Paper III. The second element concerns the description of interparticleforces, which govern colloidal stability and, by extension, deposition. This element thusrepresents the theoretical foundation for Paper IV.

3.1 Equilibrium shapes of nanoparticlesThe thermodynamic equilibrium shape of a NP can be determined by means of a Wulffconstruction [57]. Consider a particle of a crystalline material 𝑀 consisting of facets 𝑖with surface normals 𝑛𝑖 with associated miller indices {(ℎ𝑖𝑘𝑖𝑙𝑖)}𝑖=1,2,…, surface free ener-gies 𝛾𝑖 ≡ 𝛾[𝑀(ℎ𝑖𝑘𝑖𝑙𝑖)], areas 𝐴𝑖, and distances to the crystal center ℎ𝑖. The total volumeof the particle is given by

𝑉 = ∑𝑖

13ℎ𝑖𝐴𝑖. (3.1)

The total free energy of the particle can be decomposed into contributions from thebulk and the surface

𝐺 = 𝐺bulk [𝑀] +∑𝑖𝛾𝑖𝐴𝑖. (3.2)

¹The curious reader is referred to the review by Barnard[56].

15

Chapter 3. Elements of nanoparticle modeling

The equilibrium shape of a particle can be found be minimizing 𝐺 under the constraintof constant volume, introduced through a Lagrange multiplier [58],

𝛿(𝐺 − 𝜆𝑉 ) = 𝛿𝐺bulk [𝑀] +∑𝑖𝛿 (𝛾𝑖𝐴𝑖 −

𝜆3ℎ𝑖𝐴𝑖) = 0. (3.3)

In this equation, the first term vanishes since the volume is fixed and consequently thevariation can only vanish if the terms in the summation independently vanish. Thisproves the Gibbs-Wulff theorem

ℎ𝑖 = constant × 𝛾𝑖. (3.4)

This equation tells us that the equilibrium shape, or Wulff shape, is given by the set ofpoints

𝒲 = {𝒙 ∶ 𝒙 ⋅ 𝒏𝑖 ≤ 𝛾 [𝒏𝑖] for all 𝒏𝑖} . (3.5)

The term Wulff construction is often used interchangeably with Wulff shape, and weshall make no distinction between the two in this text.

3.2 A thermodynamic model of adsorptionOn an atomic level, adsorption refers to a processwhere any number atoms ormolecules,the adsorbates, becomes bonded to a surface, the adsorbent. Adsorption is often catego-rized as either chemisorption when a chemical bond is formed between adsorbate andsurface, or physisorption when the adsorbate is weakly bonded to the surface throughvan der Waals interactions. The goal of this section is to establish a thermodynamicmodel of adsorption that leads to a computable expression for the free energy. Thisenables e.g., prediction of the phase stability of a given surface system as a function ofits environment, which, through the Wulff construction described above, also includesNPs. A pivotal role is played by atomistic simulation methods, which provide the totalenergies needed as input to the model.

Consider the interfacial region between an elemental crystal ² 𝑀 and a source ofpotential adsorbate molecules or atoms 𝑋 . The surface 𝑀(ℎ𝑘𝑙) is assumed to be flat,single crystalline and initially without defects. The nature of the source, includingphase, composition and properties such as dilution, is left unspecified for the moment.To account for adsorption, we write the surface free energy in Eq. (3.5) as

𝛾 [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] = 𝛾 [𝑀(ℎ𝑘𝑙)] + Δ𝛾 [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] , (3.6)

where the first term on the right hand side is the clean surface free energy, and thesecond term represents the change in free energy induced by adsorption of 𝑁𝑋 units

²The restriction to monatomic crystals is not strictly necessary but simplifies the discussion.

16

3.2. A thermodynamic model of adsorption

of 𝑋 . We refer to this last term as the surface free energy of adsorption. To find anexpression for this unknown quantity, we can represent a non-dissociative adsorptionevent (Fig. 3.1) as a reaction ³

𝑀 (ℎ𝑘𝑙) + 𝑋 −→ 𝑀 (ℎ𝑘𝑙) ∶ 𝑋 . (3.7)

Surface

Source

M(hkl)

X

M(hkl) : X

Adsorption

Figure 3.1: Non-dissociative adsorption on the surface of a material𝑀 in contact with asource of molecules 𝑋 . This can be regarded as a reaction bringing the isolated surfaceandmolecules together in an adsorbed state. The free energy of the reaction determinesif it is endothermic or exothermic.

It is useful to generalize this reaction to allow for the possibility of a surface reconstruc-tion. In addition to structural changes, the number of𝑀 atoms in the interfacial regioncan then also undergo a change 𝑁𝑀 → 𝑁𝑀 + Δ𝑁𝑀 upon adsorption of 𝑁𝑋 molecules.This net change can be expressed asΔ𝑁𝑀 = 𝑁𝑎−𝑁𝑣 , where𝑁𝑎 is the number of adatomsand 𝑁𝑣 is the number of vacancies. The reaction is then written in stoichiometric formas

𝑁𝑀𝑀(ℎ𝑘𝑙) + 𝑁𝑋𝑋𝐴Δ𝛾−−−−→ (𝑁𝑀 + Δ𝑁𝑀 )𝑀 (ℎ𝑘𝑙) ∶𝑁𝑋𝑋 − Δ𝑁𝑀𝑀, (3.8)

where a change in free energy per area unit Δ𝛾 ≡ Δ𝛾 [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋 ] has been intro-duced that when multiplied by the area gives the total change in Gibbs free energy Δ𝐺

³In the case of dissociative adsorption when the source consists of molecules 𝑋𝑌 the dissociationmust be included in the adsorption process. For example, if 𝑋 = 𝑌 the reaction becomes𝑀(ℎ𝑘𝑙)+ 1

2𝑋2 −→𝑀(ℎ𝑘𝑙) ∶ 𝑋 .

17


associated with the reaction. The change in surface free energy is obtained by takingthe free energy difference of the right and left-hand side of Eq. (3.8) and dividing by thearea, leading to

Δ𝛾 [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] = 1𝐴 (𝐺 [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] −

𝐺 [𝑀(ℎ𝑘𝑙)] − Δ𝑁𝑀𝜇 [𝑀] − 𝑁𝑋𝜇 [𝑋]) .(3.9)

The terms on the right-hand side of Eq. (3.9) represent, going from left to right, thefree energy of the surface with adsorbates, the free energy of the clean surface, thenet change in 𝑀 atoms Δ𝑁𝑀 times the chemical potential of the bulk 𝜇 [𝑀], and thenumber of adsorbates 𝑁𝑋 times the chemical potential of the molecules 𝜇 [𝑋]. Here,the bulk and the source are regarded as reservoirs with associated chemical potentialsthat measure their propensity to exchange matter with the interfacial region.

It is often convenient to write 𝜇𝑋 relative to a standard state 𝜇∘ [𝑋],𝜇 [𝑋] = 𝜇∘ [𝑋] + 𝑘𝐵𝑇 ln 𝑎 [𝑋] . (3.10)

The definition of the adsorbate activity 𝑎 [𝑋] in the above expression, as well as thechoice of the standard state, depends on the nature of the reservoir.

3.2.1 Realization using DFTOur thermodynamic model of adsorption on surfaces, and consequently also on NPs byvirtue of Eq. (3.5), is now in principle fully defined by Eqs. (3.6), (3.9), and (3.10). Duethe complicated electronic structure of the interfacial region between a crystal and anadsorbate, a quantum mechanical treatment of the system is often necessary to obtainaccurate results. As discussed in the beginning of this chapter, DFT is the method ofchoice for such calculations. However, the energies computed using DFT correspondthe electronic energy contribution in a vacuum, and hence additional models and ap-proximations must be introduced if we are to calculate the surface free energies inEq. (3.6).

First, we shall assume that the temperature does not deviate significantly from 𝑇 𝑟 =298.15 K, i.e. room temperature. This means that vibrational contributions to the freeenergy are small and the free energy of a solid 𝑀𝑠 can be well-represented by DFT.Accordingly, we prescribe the solid an absolute free energy 𝐺[𝑀𝑠] ≈ 𝐸DFT [𝑀𝑠]. Thefirst three terms on the right hand side of Eq. (3.9) can then be accurately determinedfrom DFT calculations.

The remaining, and most challenging, problem pertains to the calculation of the ad-sorbate chemical potential appearing in Eq. (3.9). Rather than attempting a generaltreatment, we shall consider two different approaches and illustrate them for appli-cations relevant to this thesis. The first approach is to combine DFT with statistical

18


mechanics, which will be illustrated for the case of dissociative adsorption from thegas-phase, which is relevant for Paper I. The second approach is to combine DFT withexperimental data, which will be illustrated for aqueous halide adsorption, which isrelevant for Paper II and to a lesser extent Paper III.

Gas-phase adsorption: combining DFT with statistical mechanics

The dissociative adsorption of a gas-phase dimer 𝑋2 can be described by the reaction𝑀+ 1

2𝑋2 ⟶ 𝑀 ∶𝑋 and the chemical potential of𝑋 appearing in Eq. (3.9) is accordingly

given by12𝜇[𝑋2]. If we assume 𝑋2 behaves like an ideal gas, analytical expressions for

the various free energy contributions can be derived from statistical mechanics [59].By definition 𝜇[𝑋2] = 𝐻[𝑋2] − 𝑇𝑆[𝑋2]⁴ where the enthalpy can be written as

𝐻[𝑋2] = 𝐸elec[𝑋2] + 𝐸ZPE[𝑋2] + ∫𝑇

0𝐶𝑃𝑑𝑇 . (3.11)

Here, we have separated out the electronic and the ionic ZPE contribution to the to-tal energy, the former will be calculated using DFT. The heat capacity in the integralin Eq. (3.11) receives contributions from translational, rotational, vibrational, and elec-tronic degrees of freedom, although the electronic contribution is typically neglected.For brevity the explicit form of these terms is not given; except for the vibrational terms,which require that the ionic vibrational energies of the system are known, they can beimmediately evaluated. Additionally, as we shall see, the vibrational part of the fulladsorption free energy can often be neglected, in which case no vibrational calculationis needed. The same considerations apply to the entropy, which we write relative to itsvalue at a reference pressure conventionally chosen as 𝑝∘ = 1 bar as

𝑆[𝑋2](𝑇 , 𝑝) = −𝑘𝐵 log ( 𝑝𝑝∘) + 𝑆∘[𝑋2](𝑇 , 𝑝∘)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=𝑆trans+𝑆rot+𝑆vib(3.12)

Putting Eqs. (3.11) and (3.12) together, the chemical potential can be brought into theform

𝜇[𝑋2](𝑇 , 𝑝) = 𝐸elec[𝑋2] + 𝜇∘vib[𝑋2](𝑇 ) + 𝜇∘trans,rot[𝑋2](𝑇 ) − 𝑘𝐵𝑇 log ( 𝑝𝑝∘) . (3.13)

Here, the vibrational contributions, including the ZPE, have been grouped together in𝜇∘vib[𝑋2]. The remaining contributions at reference pressure, which mainly consists ofa translational and rotational component, are collected in 𝜇∘trans,rot[𝑋2]. We are now in

⁴Note the use of the commonly applied “lazy” notation for thermodynamic quantities 𝑄 where 𝑄and 𝜕𝑄/𝜕𝑁 are both written with capital letters and the right form has to be inferred from the context.

19


a position to consider the full surface free energy of adsorption given in Eq. (3.9) again,where for simplicity it assumed that no reconstruction of surface takes place. We nowmake the approximation

𝐴Δ𝛾 [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] =𝐺[𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] − 𝐺[𝑀(ℎ𝑘𝑙)] − 𝑁𝑋12𝜇[𝑋2]

≈𝐸DFT[𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋] − 𝐸DFT[𝑀(ℎ𝑘𝑙)]−12 (𝐸DFT[𝑋2] + 𝜇∘trans,rot[𝑋2](𝑇 ) − 𝑘𝐵𝑇 log ( 𝑝

𝑝∘)) ,(3.14)

the validity of which relies on error-cancellation of similar terms between the compos-ite surface-adsorbate system as well as the clean surface and gas-phase molecules takentogether. For instance, it can be argued that the difference in vibrational free energybetween a surface with adsorbates and the clean version of that surface plus gaseousadsorbates is small [60].

Equation (3.14) provides us with a way of predicting the stability of gas-phase adsor-bate structures on any crystalline surface using only 0 K DFT total energies as input,yet depends on both temperature and pressure through the two last terms.

Aqueous ion adsorption: combining DFT with experimental data

Aqueous ions are one of the more challenging types of adsorbates to describe accu-rately. The additional interactions and degrees of freedom introduced by the presenceof the solvent make a treatment in terms of statistical mechanics difficult and causeperformance issues if explicitly included in DFT calculations. This section delineatesan alternative approach based on a combination of experimental data with DFT calcu-lations. The advantage of this approach is that it allows us to completely avoid the useof DFT for solvated and ionized adsorbate states. The key idea is to find a referencebulk system that is well-described by DFT and contains the ion of interest and thenuse experimental free energies to connect it to the desired adsorbate state. This typeof consistent combining scheme was introduced by Persson et. al., who focused on thedescription of metal oxides [61]. The scheme presented here is, in essence, a variationof their approach. While a detailed description is given for the case of aqueous halideadsorption, extension to other ionic species is straightforward.

Experimental free energies are tabulated for a standard state under reference con-ditions and thus correspond to the first term in Eq. (3.10). Here, the comprehensivethermodynamic data set procured and corrected by Hunenberger and Reif [62] will beused. The conventional standard state is that of an ideal molar solution under refer-ence conditions 𝑃 ∘ = 1 bar and 𝑐∘ = 1M, and the ∘-superscript will be used to denotevalues at these conditions. Note that standard values of thermodynamic quantities canstill be a function of temperature, which applies in particular to 𝜇∘, while experimental

20


values are often tabulated only at room temperature 𝑇 𝑟 = 298.15 K. Accordingly, when(𝑐, 𝑃 , 𝑇 ) = (𝑐∘, 𝑃 ∘, 𝑇 𝑟 ), a •-superscript will be used instead.

Now we let 𝑋−aq for 𝑋 ∈ {F,Cl,Br, I}, and assume that the halide solution is ideal,

which means that the activity coefficient Υ is unity. The activity on the right-hand sideof Eq. (3.10) is then reduced to

𝑎 [𝑋−aq] = Υ [𝑋−

aq]𝑐 [𝑋−

aq]𝑐∘ ≈ 𝑐 [𝑋−

aq]𝑐∘ . (3.15)

The notation used in this section closely follows that of Hunenberger and Reif [62]and is succinctly summarized in Fig. 3.2. Note the distinction between reaction andformation energies. The former are free energy differences between two states, e.g.,gaseous ion and solvated ion, and denoted Δ𝑟𝐺 while the latter measure the differencebetween a state and the elemental state of the halogen and are denoted Δ𝑓𝐺. In thiscontext, the elemental state refers to the naturally occurring state of the ion, which forthe halogens is

12𝑋el = F2(g), Cl2(g), Br2(l), I2(s).

The task is now to connect the experimental formation reaction energies to the totalDFT energies. As we have previously argued, the energy of solids are well-described byDFT under, or close to, ambient conditions. Consequently, we can assign DFT energiesto the absolute enthalpies of an alkali-halide salt𝑀𝑋𝑠 and the corresponding pure alkalimetal. The absolute enthalpy of a halogen 𝑋 in its elemental state can now be fixed ifwe insist that the reactions in Fig. 3.2 all proceed with the correct experimental reactionenergies

𝐻 • [𝑋el] = 𝐸DFT [NaXs] − 𝐸DFT [Nas] − Δ𝑓𝐻 •exp [NaX𝑠] . (3.16)

The absolute enthalpy of an aqueous halide ion is then simply obtained as

𝐻 • [𝑋−aq] = 𝐻 • [𝑋el] + Δ𝑓𝐻 •

exp [𝑋−aq] . (3.17)

The standard chemical potential can now be approximated according to

𝜇∘ [𝑋−aq] = 𝐻 ∘ [𝑋−

aq] (𝑇 ) − 𝑇𝑆∘exp [𝑋−aq] (𝑇 ) ≈ 𝐻 • [𝑋−

aq] − 𝑇𝑆•exp [𝑋−aq] . (3.18)

The approximation here consists of neglecting the individual functional dependenciesof𝐻 • and 𝑆•exp on temperature, however, an exact equality holds when 𝑇 = 𝑇 𝑟 . It followsfrom Eqs. (3.10), (3.15), and (3.17) that the chemical potential at any concentration canbe calculated as

𝜇 [𝑋−aq] = 𝜇∘ [𝑋−

aq] + 𝑘𝐵𝑇 log (𝑐 [𝑋−aq]

𝑐∘ ) . (3.19)

21


atomization

ionization

Elemental state

Atomized gas

Ionized gasSalt crystal Aqueous solution

dissolution

Crystal Gas/liquid

rectic

ulation solvation

Figure 3.2: Notation for free energies and reactions relating to the formation and disso-lution of alkali-halide salts. The alkali species are denoted by𝑀 and the halogens by 𝑋 .In their elemental, i.e. naturally occurring state, the formation energy of a species 𝑀or 𝑋 is by definition zero. Formation energies for all other states can then be measuredrelative to the elemental state and denoted Δ𝑓𝐺. In going from the elemental state to afinal state, we can pass through one or more intermediate states which are connectedby reaction energies Δ𝑟𝐺.

The final expression for the surface free energy of adsorption, see Eq. (3.14), of a halideas approximated using our combining scheme is

Δ𝛾 [𝑀(ℎ𝑘𝑙) ∶ (𝑁𝑋𝑋−aq)] ≈

1𝐴 (𝐸DFT [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋𝑋]−

𝐸DFT [𝑀(ℎ𝑘𝑙)] − 𝑁𝑋 (Φ [𝑀(ℎ𝑘𝑙)] + 𝜇 [𝑋−aq])) .

(3.20)

Here, the DFT calculation for a charged slab implied by the first term on the right inEq. (3.9) is avoided and replacedwith the energy of a neutral system 𝐸DFT [𝑀(ℎ𝑘𝑙) ∶ 𝑁𝑋]using a thermodynamic cycle [63] that implies subtracting an appropriate multiple ofthe work function Φ [𝑀(ℎ𝑘𝑙)] of the clean surface.

It is important to note that, since we want to avoid explicit inclusion of water inthe DFT calculations, all interactions between surface and solvent are neglected in thetwo first terms on the right-hand side of Eq. (3.20). In Paper II, the effect of neglectingthese surface-solvent interactions is investigated by comparing the results obtainedfrom conventional DFT without and with an implicit solvent model [64, 65].

22

3.3. Interparticle forces and DLVO theory

3.3 Interparticle forces and DLVO theoryThis section presents another element of modeling colloidal NPs, namely that of theforces acting between NPs themselves, rather than between the atoms on the surfaceof a single NP as in the previous sections of this chapter. A closely related concept isthat of colloidal stability. In the context of colloidal science this term is not used in thethermodynamic sense of a system state that minimizes the appropriate thermodynamicpotential.⁵ Rather, it refers to a thermodynamically metastable state in which the sys-tem has become kinetically trapped. In this state the discrete phase remains dispersedin the continuous phase without settling or precipitating, a condition which requiresthe particles to obey Brownian dynamics. In practice, most colloids do settle and reachthermodynamic equilibrium over time. The rate of this process can vary dramatically,milk settles over a span of a few weeks while sols can remain intact for years.

Restricting the discussion to the case where the discrete phase consists of particleswith mass 𝑚, a limiting size for the Brownian regime can be obtained [66]. Considerthe distribution of particles at height ℎ in a continuous medium of density 𝜌𝑚. In equi-librium this distribution follows Boltzmann statistics with weights determined by thegravitational potential energy

𝑃(ℎ) ∝ exp (−𝑚𝑔ℎ𝑘𝐵𝑇). (3.21)

Accordingly, the distribution has a mean value ⟨ℎ⟩ = 𝑘𝐵𝑇/𝑚𝑔. Assuming the particlesare spherical with radius 𝑎 and density 𝜌, they are suspended if ⟨ℎ⟩ > 2𝑎, which solvingfor 𝑎 and adjusting the density to include buoyancy yields

𝑎 < ( 3𝑘𝐵𝑇8𝜋𝑔(𝜌 − 𝜌𝑚)

)1/4

. (3.22)

For, e.g., gold particles in water under ambient conditions this requirement yields anupper limit of 𝑎 ≲ 0.5 𝜇m. Note, however, that even if a suspension initially containsonly particles of dimensions consistent with Eq. (3.21), the criterion may still be vio-lated. This occurs if the attractive interparticle forces in the systems are stronger thantheir repulsive counterparts, in which case neighboring particle adhere to each otherand coalesce or form aggregates that settle when they reach the limiting size.

The twomost important types of interactions in colloids are vdW interactions, whichare attractive, and EDL interactions, which can be either repulsive or attractive de-pending on the nature of the colloid. In DLVO theory, which is historically the mostsuccessful theory of colloidal stability, the total force between two colloidal particles issimply taken as the sum of the forces derived from the vdW and EDL interactions. In

⁵This would imply a phase separation between the discrete and continuous phases of the colloid.

23


the next two sections, we consider each of these two contributions separately and thendescribe in Sect. 3.3.3 how approximations for the combined DLVO interaction energyof convex bodies can be found.

3.3.1 van der Waals interactionsThe driving force behind adhesion of colloidal particles comes from dispersive inter-actions, which are long-ranged, attractive, and present in all atomic systems. Theseinteractions can, at least partially, be understood in terms of a semi-classical picture:quantum mechanical charge fluctuations create an instantaneous dipole in an atom 𝐴,which subsequently induces an aligned, instantaneous dipole in a neighboring atom 𝐵.The result is an interaction between the two induced dipoles𝐴𝐵 that is always favorableand decays as 1/𝑟6. The true origin of dispersive interactions is, however, purely quan-tummechanical and the reader is referred to Appendix A for amore detailed discussion.Dispersive interactions are often referred to as van-der-Waals (vdW) interactions, al-though some authors use this term to also include dipole-induced dipole (Debye) anddipole-dipole (Keesom) interactions. In contrast to the dispersive interactions, the De-bye and Keesom interactions can be adequately described by classical physics [67]. Inthis section the terms vdW and dispersive interactions are used synonymously.

The total vdW energy resulting from two interacting bodies ℬ1,ℬ2, assuming thatthe microscopic contribution from an interacting atomic pair follows the asymptotic1/𝑅6 form, is given by

𝑈 vdw = − ∑𝐼 ∈ℬ1

∑𝐽∈ℬ2

𝐶 𝐼 𝐽6𝑅6𝐼 𝐽

, (3.23)

where 𝐶 𝐼 𝐽6 are coefficients that determine the strength of the interaction between twoatoms 𝐼 , 𝐽 at a fixed separation 𝑅𝐼 𝐽 . In the limit of large bodies, a continuum treatmentof the vdW interaction allows for the derivation of closed-form analytic expression forthe total energy in a few important cases. We assume that the bodies are composedof the same atomic building blocks with number density 𝜌 and interaction coefficient𝐶6. In the continuum limit Eq. (3.23) can then be cast as a double integral over the twobodies,

𝑈 vdw = −∫ℬ1𝑑𝑹1 ∫ℬ2

𝑑𝑹2𝜌2𝐶6

|𝑹1 − 𝑹2|6. (3.24)

There are three important special cases for which the above integration can be carriedout analytically to obtain a closed-form expression for the interaction energy. Thefirst and simplest case is that of two infinite flat plates separated by a distance 𝑑 . Theinteraction energy per unit area is then given by

𝑢vdw𝑓 (𝑑) = − 𝐴𝐻12𝜋𝑑2 . (3.25)

24


Here, 𝐴ℎ = 𝜋2𝜌2𝐶6 is known as the Hamaker constant, in honor of H. C. Hamaker whowas among the first to explore dispersive interactions between macroscopic bodies.Next we consider two interacting spheres, for which Hamaker famously derived [68]

𝑈 vdw𝑠 = − 𝐴ℎ6 ( 2𝑎1𝑎2

𝑑2 + 2𝑑(𝑎1 + 𝑎2)+ 2𝑎1𝑎2𝑑2 + 2(𝑎1 + 𝑎2) + 4𝑎1𝑎2

+

log ( 𝑑2 + 2𝑑(𝑎1 + 𝑎2)𝑑2 + 2(𝑎1 + 𝑎2) + 4𝑎1𝑎2

)) ,(3.26)

for spheres of radii 𝑎1, 𝑎2 and shortest separation 𝑑 . If we let the radius of one of thespheres tend to infinity an expression for the interaction of a single sphere with radius𝑎 with an infinite flat surface is obtained as a bonus

𝑈 vdw𝑠𝑓 (𝑑) = −𝐴𝐻

6 (𝑎𝑑 + 𝑎𝑑 + 2𝑎 + log ( 𝑑

𝑑 + 2𝑎)) . (3.27)

3.3.2 Electrostatic interactionsIn order to stabilize the colloidal particles against aggregation or coalescence the netinter-particle forces must be repulsive. One way to stabilize the particles is via so-called electrostatic stabilization⁶, where the surface charge of the particles is tuned sothat they repel each other.

There are many pathways through which a particle surface can acquire a net charge.These include, for instance, adsorption of a charged species, dissociation of a surfacespecies into charged fragments as well as the build-up of a positive or negative excessof electrons at the surface. Once the surface has become charged, oppositely chargedspecies are attracted towards it forming an electric double layer.

Limiting the discussion to colloidal particles in ionic solutions, which is the settingrelevant for wet-chemical synthesis processes, an illustrative derivation of the electro-static interaction between two charged spherical particles is provided below. Detailedaccounts can be found in elementary textbooks on electrolyte solutions [69]. The dou-ble layer charge distribution gives rise to an electric surface potential 𝜙 that can beobtained from the Poisson equation

Δ𝜙 (𝒓) = −𝜌(𝒓). (3.28)

In equilibrium, the concentrations of each ionic species 𝑗 with bulk concentration 𝑛0𝑗and charge 𝑞𝑗 follow Boltzmann statistics. Combined with Eq. (3.28) this results in the

⁶Another option is steric stabilization, where polymers are grafted onto the particle surface. As twoparticles approach each other, an overlap region is created in which the confinement of the polymersleads to a loss of configurational entropy that manifests as a repulsive force.

25


PB equation𝑑2𝜙𝑑𝑥2 = − 1

𝜖0𝜖𝑟∑𝑗𝑞𝑗𝑐∞𝑗 𝑒−𝑞𝑗𝜙/(𝑘𝐵𝑇 ), (3.29)

where 𝜖 = 𝜖0𝜖𝑟 and 𝜖𝑟 is the dielectric constant of the medium. Since the electric poten-tial also appears in the Boltzmann factors in Eq. (3.29), the PB equation is non-linearand no closed-form analytic solutions are available. A common approximation is theDebye-Hückel linearization, obtained by Taylor expanding the Boltzmann factors to sec-ond order yielding the LPBE

Δ𝜙 (𝒓) = 𝜅2𝜙 (𝒓) . (3.30)

Here, the Debye screening length

𝜅−1 =√

𝜖𝑘𝐵𝑇∑𝑗 𝑛0𝑗 𝑞2𝑗

, (3.31)

was introduced, which provides a measure of the effective range of the electrostatic in-teraction in an electrolyte. The LPBE can be solved exactly for some simple systems, themost important example of which is an electrolyte surrounded by two infinite parallelplates at separation 𝑑 . Hogg, Healy, and Fuerstenau [70] showed, assuming constantsurface potentials 𝜙1 and 𝜙2 on the plates, that an analytic expression can be derivednot only for the electrostatic potential, but also for the total interaction energy per unitarea

𝑢edl𝑓 (𝑑) = 𝜖0𝜖𝑟𝜅2 (𝜙21 + 𝜙22) (1 − coth(𝜅𝑑) + 2𝜙1𝜙2

𝜙21 + 𝜙22cosech(𝜅𝑑)) . (3.32)

This is often referred to as the Hogg-Healy-Fuerstenau formula [70]. The reason thatthe two-plate system is of special importance is because it allows for the derivationof approximate interaction expressions for arbitrary convex geometries using the Der-jaguin approximation, which will be discussed in the next section. First, we remark thatthe error in the electric potential introduced as a result of using the LPBE is 𝒪(𝜙3) forsymmetric electrolytes, i.e. if the valences of the anions and cations are equal 𝑧1 = 𝑧2,but 𝒪(𝜙2) if the electrolyte is asymmetric (𝑧1 ≠ 𝑧2). To see the significance of this,we observe that from Eq. (3.29) the range of validity for the linearization is 𝑞𝜙 ≪ 𝑘𝐵𝑇 ,which for monovalent ions in room temperature translates to 𝜙 ≪ 25mV. Now, forsymmetric electrolytes it has been noted that the LPBE yields decent results even if𝜙 ≈ 25mV [70], but due to the increased approximation error the same cannot be saidfor the asymmetric case. Hence, in practice one is often forced to solve the non-linearPB equation for asymmetric electrolytes, which is the case, e.g., in Paper IV.

Even when the LPBE is not valid, however, it is still straightforward to obtain thetwo-plate interaction energy if the PB equation can be solved numerically. In this case,

26


we note that the disjoining-pressure Π of the plates is given by the sum of the osmoticand electromagnetic pressures according to

Π(𝑑) = 𝑘𝐵𝑇 ∑𝑗𝑐0𝑗 (𝑒−𝑞𝑗𝜙𝑑/(𝑘𝐵𝑇 ) − 1) − 𝜖0𝜖𝑟

2𝑑2𝜙𝑑𝑑𝑥2 . (3.33)

The total total interaction energy can subsequently be obtained by integration

𝑢edl𝑓 (𝑑) = ∫∞

𝑠=𝑑Π(𝑠)𝑑𝑠. (3.34)

3.3.3 Surface element integrationAs alluded to in the previous section, theDerjaguin approximation can be used to obtainthe total DLVO interaction energy between two arbitrary convex bodies, given that theinteraction energy for two corresponding infinite, parallel plates with the same bound-ary conditions can be calculated at any separation. More precisely, given the distanceof closest separation 𝐷 of two arbitrary convex bodiesℬ1,ℬ2, can be approximated as

𝑈 (𝐷) = 𝑔(𝜿1, 𝜿2) ∫∞

𝑠=𝐷𝑢tot𝑓 (𝑠)𝑑𝑠 (3.35)

where 𝑢tot𝑓 = 𝑢vdw𝑓 +𝑢edl𝑓 is the total DLVO energy of the two-plate system (see Eqs. (3.25)and (3.34)) and 𝑔 is a geometric factor that depends on the principal curvatures 𝜿1, 𝜿2of the two surfaces 𝒮1,2 = 𝜕ℬ1,2. Despite the usefulness of the Derjaguin approxima-tion in finding analytic expressions for the DLVO interaction between a wide rangeof important geometries [67], it suffers from critical limitations. More precisely, thedistance of closest approach and characteristic range of the DLVO interaction must besmall compared to the radii of curvature of both the surfaces [71].

For the specific case of a convex body interacting with an infinite flat plate, a moreaccurate method of obtaining the interaction energy was proposed by Bhattacharjeeand Elimelech [72] termed SEI. Indeed, it was shown that using SEI, the exact vdWinteraction (Eq. (3.27)) was recovered and that the EDL interaction calculated basedon the Hogg-Healy-Fuerstenau solution Eq. (3.32) matched results from finite elementsolutions of the PB equation within the linear regime.

To formulate the SEI method, we consider an infinite flat plate𝒜 and a convex body𝒮 , with corresponding surface elements 𝑑𝑨 = 𝒌𝑑𝐴 and 𝑑𝑺 = 𝒏𝑑𝑆, where 𝒌 and 𝒏are the respective surface normals. Contributions to the total interaction energy arethen computed for each surface element of 𝒮 as 𝑑𝑈 = 𝑑𝑺 ⋅ 𝒌𝑢𝑓 (𝑠), where 𝑠 is the localseparation between 𝑑𝑺 and 𝒜 . Accordingly, the total interaction energy is

𝑈 (𝐷) = ∫𝒮 𝑑𝑺 ⋅ 𝒌𝑢tot𝑓 (𝑠) = ∫𝒜 𝑑𝐴 𝒏 ⋅ 𝒌|𝒏 ⋅ 𝒌|𝑢

tot𝑓 (𝑠) (3.36)

27


where 𝐷 is the distance of closest approach between 𝒮 and𝒜 . Subsequent work by thesame authors extended the SEI method to the more general case of two convex bodies[73], and while it still constitutes an improvement over the Derjaguin approximation,the predicted energies are in this more general case only accurate when the charac-teristic interaction range is short. For instance, the exact Hamaker expression for thevdW interaction between two spheres given in Eq. (3.26) is not reproduced but ratherthe SEI value tend towards the exact value as the ratio between the radii increases, i.e.when approaching the sphere-plate limit. The SEI method is the basis of the modelimplemented in Paper IV to evaluate the efficacy of pre-patterned Ni structures on aSiO2 substrate in attracting citrate-stabilized Au NPs.

28

4First-principles calculations

”But, gentlemen, that’s not physics.”

Ceylonese parrot nominated as chairman for all seminars onquatum mechanics in Göttingen 1927.

All properties of a system of interacting, non-relativistic particles are in principledetermined by its many-body state |Φ(𝑡)⟩, which is obtained as a solution to the time-dependent Schrödinger equation

𝑖ℏ 𝜕𝜕𝑡 |Φ(𝑡)⟩ = 𝐻 |Φ(𝑡)⟩ , (4.1)

where𝐻 is the Hamilton operator of the system. The spectrum of this operator, definedby the eigenvalue equation

𝐻 |Ψ⟩ = 𝐸 |Ψ⟩ , (4.2)

corresponds to the allowed energies 𝐸 of the system. Equation (4.2) is often referredto as the time-independent Schrödinger equation. The eigenstates |Ψ⟩ may be used asa basis for an expansion of an arbitrary many-body state, but for many applications itis sufficient to know the ground state of the system i.e. the lowest eigenvalue 𝐸0 of 𝐻and its associated eigenstate |Ψ0⟩.

For a system consisting of 𝑁 electrons and 𝑀 atomic nuclei, the position-space rep-resentation of the Hamilton operator in Hartree atomic units (ℏ = 𝑒 = 𝑚𝑒 = 4𝜋𝜖0 = 1)is given by

29

Chapter 4. First-principles calculations

𝐻 =

𝑇⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞− 12

𝑁∑𝑖∇2𝑖 +

𝑉int⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞𝑁∑𝑖=1

𝑁∑𝑗>𝑖

1||𝒓𝑖 − 𝒓𝑗 ||

𝑉ext⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞−

𝑁∑𝑖=1

𝑀∑𝐼=1

𝑍𝑖|𝒓𝑖 − 𝑹𝐼 |

− 12

𝑀∑𝐼=1

∇2𝐼𝑀𝐼⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑇ion

+𝑀∑𝐼=1

𝑀∑𝐽>𝐼

𝑍𝐼𝑍𝐽||𝑹𝐼 − 𝑹𝐽 ||⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝐸nn

(4.3)

where 𝑀𝐼 , 𝑍𝐼 , 𝑹𝐼 denote the mass, charge and position of the nuclei, respectively, and𝒓𝑖 are the positions of the electrons. Exact solutions to Eq. (4.2) are only possible for1-body systems or 2-body systems that can be reduced into 1-body problems, as is thecase of an isolated hydrogen atom. In all other cases, we must resort to approximationsand numerical solutions. If the approximations giving rise to a certain method are ofsuch nature that the resulting equations are free from adjustable physical parameters,the method is customarily referred to as an ab initio or first-principles method.

A common approximation for manymethods seeking to solve Eq. (4.2) for the Hamil-ton operator in Eq. (4.3) is to make use of the fact that the nuclei are much heavier thanthe electrons. For the electrons this means that they see the nuclei as charges frozen intheir instantaneous location, hence the nuclear kinetic energy term can be dropped andthe ion-ion repulsion 𝐸𝑛𝑛 simply amounts to a constant shift of the Hamilton operator.This is known as the Born-Oppenheimer approximation [74] and allows us to write asimplified electronic Hamilton operator

𝐻e = 𝑇 + 𝑉int + 𝑉ext. (4.4)

With the spin variables suppressed, the time-independent Schrödinger equation for theelectrons now reads

𝐻𝑒Ψ𝑒({𝒓𝑖}; {𝑹𝐼 }) = 𝐸𝑒Ψ𝑒({𝒓𝑖}; {𝑹𝐼 }) (4.5)

and the total energy is 𝐸tot = 𝐸𝑒 + 𝐸𝑛𝑛. Similarly, we can write a separate Schrödingerequation for the nuclei, which can then be regarded asmoving in amean-field generatedby the electrons. Here, a full quantum mechanical treatment is often not necessaryand the nuclei can be regarded as classical particles moving in the potential 𝐸tot({𝑹𝐼 })obtained by solving the electronic problem Eq. (4.5) and often referred to as the potentialenergy surface. The force on a nuclei 𝐼 is given by

𝑭𝐼 = − 𝜕𝜕𝑹𝐼

𝐸tot({𝑹𝐼 }) = −⟨Ψ |||𝜕𝐻𝜕𝑹𝐼

||| Ψ⟩ − 𝐸 𝜕𝜕𝑹𝐼

⟨Ψ|Ψ⟩⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=0 if basis is complete

. (4.6)

30

4.1. Density functional theory

This is known as the Hellmann-Feynman theorem and evaluation of the first term inEq. (4.6) shows that the forces on the ions are simply the electrostatic forces due tothe electronic charge distribution. Numerical solutions to the Schrödinger equationare typically based on a basis expansion of the wavefunction (Sect. 4.1.5), however, andonly a finite number of such functions can be included. In this case, the derivativein the second term in Eq. (4.6) is not zero if the basis functions depend on the ionicpositions. As a result the ions experience so-called Pulay forces that need to be correctedfor according to Eq. (4.6).

Despite greatly simplifying the original problem, Eq. (4.4) further theoretical devel-opment is required to obtain equations can be solved, at least numerically.

4.1 Density functional theoryDensity functional theory (DFT) is one of the cornerstones of modern computationalmaterials physics and provides a way of solving the Schrödinger equation to a highdegree of accuracy with an efficiency that makes calculations involving several hun-dreds of atoms tractable on modern supercomputers. As alluded to by the name, theelectronic density plays a fundamental role in this theory and is defined in terms of themany-body electronic wavefunction from Eq. (4.4) as

𝑛(𝒓) = ∫ 𝑑𝒓2…𝑑𝒓𝑁 |Ψ(𝒓, 𝒓2, … , 𝒓𝑁 )|2 (4.7)

where the subscript has been dropped from Ψ𝑒 and the parametric dependence on thenuclear coordinates has been suppressed.

4.1.1 The Hohenberg-Kohn theoremsThe theoretical foundation of DFT rests on the shoulders of two theorems proved byHohenberg and Kohn (HK) in 1964 [75]. We shall be content with giving a concise sum-mary of them here, for a more detailed account readers are referred to one the manytextbooks covering the topic [76, 77]. For a stationary, interacting, non-degeneratesystem of electrons there is a one-to-one mapping between 𝑉ext, |Ψ0⟩ and 𝑛(𝒓). Conse-quently, the ground state is a functional of the ground state density, which is denoted|Ψ[𝑛]⟩. This functional is unique and takes the same form regardless of 𝑉ext, providedthat 𝑉int does not change. This implies in turn that the ground state expectation valueof any observable is also a functional of the density

𝑂[𝑛] = ⟨Ψ[𝑛]| 𝑂 |Ψ[𝑛]⟩ . (4.8)

Of particular interest is the ground state energy

31


𝐸[𝑛] = ⟨Ψ[𝑛]| 𝐻𝑒 |Ψ[𝑛]⟩ = ⟨Ψ[𝑛]| 𝑇 + 𝑉int |Ψ[𝑛]⟩ + 𝐸ext (4.9)

where the the external energy functional

𝐸ext[𝑛] = ∫ 𝑑𝒓 𝑛(𝒓)𝑉ext(𝒓). (4.10)

has been defined. The total energy functional in Eq. (4.9) can be shown to obey avariational principle with a minimum corresponding to the true ground state density,i.e.

𝐸[𝑛] < 𝐸[𝑛′] , ∀𝑛′ ≠ 𝑛 (4.11)

An appealing feature of these results is that the unwieldy many-body wavefunction inbypassed favor of the electronic ground state density, a scalar function of only threevariables. Note however, that the HK theorems are pure existence theorems and do notcontain a recipe for constructing an explicit expression for 𝐸[𝑛] that would allow fordetermination the ground-state energy and other observables.

4.1.2 Kohn-Sham theoryKohn-Sham (KS) theory provides us with a way of leveraging the Hohenberg-Kohntheorems to derive a practical scheme for determining the ground state density of asystem [78]. It rests on the non-trivial assumption that a system of interacting electronscan be mapped onto a system of fictitious, non-interacting electrons in such a way thatthe ground-state electron density is the same for the two systems. This new system istypically referred to as the KS auxiliary system and its existence has not been provedfor the general case; rather it is typically justified from a practical perspective by theremarkable success of KS theory in predicting many material properties. Since the KSsystem is non-interacting, it can be described by a set of single-particle, Schrödinger-like equations

[−12∇2 + 𝑉KS(𝒓)]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟≡𝐻KS

Ψ𝑖(𝒓) = 𝜖𝑖Ψ𝑖(𝒓). (4.12)

The density is then simply 𝑛(𝒓) = ∑𝑖 𝑓𝑖|Ψ𝑖(𝒓)|2, where 𝑓𝑖 is the occupation number oforbital 𝑖. Here, 𝑉KS is an effective potential for which a useful expression can be derivedby decomposing the energy functional for the KS system according to

𝐸KS = 𝑇𝑠[𝑛] + 𝐸H[𝑛] + 𝐸xc[𝑛] + 𝐸ext[𝑛] (4.13)

where the very first term is the kinetic energy functional

32


𝑇𝑠[𝑛] = −12 ∑𝑖𝑓𝑖Ψ𝑖(𝒓)∇2Ψ𝑖(𝒓). (4.14)

The interactions of the original many-body problem, other than those coming from theexternal potential, are reflected by the interacting KS density in the other two terms𝐸𝐻 and 𝐸xc. The first of these terms is the Hartree energy functional

𝐸𝐻 [𝑛] =12 ∫ 𝑑𝒓 𝑑𝒓′ 𝑛(𝒓)𝑛(𝒓

′)|𝒓 − 𝒓′| . (4.15)

The last term 𝐸xc in Eq. (4.13) is known as the exchange-correlation functional and isdefined to contain all many-body effects such that the ground state density of the KSsystem will indeed be equal to that of the original interacting system. By using theHohenberg-Kohn variational principle for 𝐸KS, a set of independent electron equationsof the form Eq. (4.12) is obtained with

𝑉KS = ∫ 𝑑𝒓′ 𝑛(𝒓′)|𝒓 − 𝒓′|⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≡𝑉𝐻 (𝒓)

+ 𝛿𝐸xc𝛿𝑛(𝒓)⏟≡𝑉xc(𝒓)

+𝑉ext(𝒓). (4.16)

Once the KS equations in (4.12) have been solved the ground state energy can be cal-culated as

𝐸0 = ∑𝑖𝑓𝑖𝜖𝑖 − 𝐸𝐻 [𝑛(𝒓)] − ∫ 𝑑𝒓 𝑛(𝒓)𝑉xc(𝒓) + 𝐸xc[𝑛]. (4.17)

4.1.3 Approximating the exchange-correlation functionalThe ground state energy as calculated from the KS system as in Eq. (4.17) is exact pro-vided that exact expressions for all terms in Eq. (4.16) are known. This is, however,not the case for the exchange-correlation functional 𝐸xc, which was defined to captureall the many-body effects not included by the other terms. 𝐸xc constitutes the majorsource of approximation in DFT.

Short of neglecting 𝐸xc altogether, the simplest approach we can take is the localdensity approximation (LDA), in which a differential volume 𝑑𝒓 is assumed to give acontribution to the total exchange-correlation energy equal to the energy density of ahomogeneous electron gas (HEG) having density 𝑛(𝒓) so that

𝐸LDAxc = ∫ 𝑑𝒓 𝑛(𝒓)𝜖LDA

xc [𝑛(𝒓)], (4.18)

where 𝜖LDAxc is the HEG energy density. Note that this expression is purely local in the

sense that 𝐸LDAxc is the sum of point-wise evaluated contributions. A natural extension

33


is to consider also variations in the density i.e., make use of the gradient |∇𝑛(𝒓)| andpossibly higher order derivatives, referred to as a generalized-gradient approximation(GGA)

𝐸GGAxc = ∫ 𝑑𝒓 𝑛(𝒓)𝜖GGA

xc [𝑛(𝒓), |∇𝑛(𝒓)|, |∇2𝑛(𝒓)|, …]. (4.19)

Such approximations are termed semi-local to indicate that the contributions, whilestill evaluated at a point, take the infinitesimal environment into account through theinclusion of density derivatives. A popular GGA-type method is the Perdew-Burke-Ernzerhof (PBE) functional [79], which takes into account first-order variations of thedensity.

4.1.4 Dispersion-correctionsAmajor shortcoming of local or semi-local exchange-correlation functional is that theydo not account for dispersive interactions. These interactions are inherently non-localand thus connect the electron density at different points in space. Therefore, from thefunctional forms Eqs. (4.18) and (4.19) of LDA and GGA-type functionals it is apparentthat they by definition cannot include non-local effects.

A simple and efficient approach to include dispersive interactions is theDFT-Dmethod[80], which in its most basic form introduces an interatomic pair potential with the cor-rect asymptotic 1/𝑟6-form to the total DFT energy 𝐸0 in Eq. (4.17),

𝐸DFT-D0 = 𝐸0 +∑𝐼 𝐽

𝐶 𝐼 𝐽6𝑅6𝐼 𝐽

. (4.20)

The 𝐶6-coefficients appearing in this equation are fitted to dispersive interactions be-tween molecules calculated using quantum-chemistry methods. There are many draw-backs to this method [81]: the 𝐶6-coefficients are agnostic to the local environment,there is no screening, only the first order correction to the vdW energy is taken intoaccount (see Appendix A), and the correct short-range vdW behavior is not reproduced.While the two latter deficiencies are addressed to some extent in later incarnations ofthe method [82, 83], it is still fundamentally semi-empirical in nature since it relieson the availability of accurate values for the 𝐶6-coefficients. For many systems, inparticular molecular ones, this level of treatment may prove sufficient, but since vdWinteraction are typically weak even small inaccuracies can yield results that are qual-itatively wrong. Several other dispersion-correction methods are available that sufferthese drawbacks to varying degrees [84–86].An ab initio approach to the problem is provided by the van der Waals density func-tional (vdW-DF) framework. The underlying theory has a long history and is quiteinvolved, hence only a brief overview is given here and the curious reader is referred

34


to more detailed accounts [87, 88]. The vdW-DF framework has given birth to a familyof functionals that can be written in the general form

𝐸vdW-DFxc [𝑛] = 𝐸GGA𝑥 [𝑛] + 𝐸LDA𝑐 [𝑛] + 𝐸nl𝑐 [𝑛]. (4.21)

Each family member is thus the sum of the exchange part of a suitably chosen GGAcompanion functional, local LDA correlation and a non-local correlation functional.The latter functional can be written

𝐸nl𝑐 [𝑛] = 12 ∫ 𝑑𝒓 𝑑𝒓′ 𝑛(𝒓)𝐾(𝒓, 𝒓′)𝑛(𝒓′) (4.22)

and accounts for the dispersive interactions. Here, the integration kernel 𝐾(𝒓, 𝒓′) con-tains information about how strongly two density regions interact depending on theirspatial separation and the asymmetry of their response. While approximations aremade in deriving Eqs. (4.21) and (4.22), no adjustable parameters with physical sig-nificance, akin to the 𝐶6 coefficients of the DFT-D method, are introduced and hencevdW-DF constitutes a genuine first-principles method. It should also be clarified thatwhile the larger vdW-DF framework accounts for many-body dispersion effects, ap-proximations made in deriving the actual functional Eq. (4.22) limit the inclusion ofsuch effects to length scales corresponding to typical binding distances. This is easilyoverlooked since Eq. (4.22) has the form of a double summation and the informationabout screening is effectively hidden in the kernel [88].

A practical problem encountered when implementing a vdW-DF functional is thatstraightforward numerical evaluation of the six-dimensional integral Eq. (4.22) is muchtoo costly. To bring the computational effort required to a manageable level, a methoddue to Román-Pérez and Soler can be employedwhere the kernel is effectively tabulatedin terms of two parameters. The non-nocal energy can subsequently be calculated byfast Fourier transforms and three-dimensional integrations [89]. This is the subject ofPaper II, which describes a reference implementation in the form of a C-library thatmake vdW-DF functionals available to an interfacing DFT code.

Two of most most prolific vdW-DF family members are vdW-DF1 [90]

𝐸vdW-DF1xc [𝑛] = 𝐸revPBE𝑥 [𝑛] + 𝐸LDA𝑐 [𝑛] + 𝐸nl(1)𝑐 [𝑛], (4.23)

and vdW-DF2 [91]

𝐸vdW-DF2xc [𝑛] = 𝐸PW86r𝑥 [𝑛] + 𝐸LDA𝑐 [𝑛] + 𝐸nl(2)𝑐 [𝑛]. (4.24)

Note that the non-local correlation functionals used above differ slightlywith regards toan underlying approximation and are consequently accompanied by different exchangefunctionals. For vdW-DF1 the conventional choice is the revised version of PBE byZhang-Yang (revPBE) [92] and for vdW-DF2 it is the refitted Perdew-Wang functional(PW86r) [93]. A more recent addition to the family is vdW-DF-cx [94]

35


𝐸vdW-DF-cxxc [𝑛] = 𝐸LV-PW86r𝑥 [𝑛] + 𝐸LDA𝑐 [𝑛] + 𝐸nl(1)𝑐 [𝑛]. (4.25)

The new feature is an exchange companion termed LV (Langreth-Vosko)-PW86r thatutilizes concepts from the vdW-DF framework also for the description of the exchangecontribution. In Paper I, the use of vdW-DF-cx was crucial in obtaining accurate resultsfor inter-chain and surface-chain vdW interactions of thiols on gold, while maintaininga good description of the surface. Indeed, it has been demonstrated that vdW-DF-cxyields excellent agreement with experimental values for many bulk properties of a largeset of solids [95].

4.1.5 Solving the Kohn-Sham equationsGiven a suitable approximation for the exchange-correlation functional the KS equa-tions (4.12) remain to be solved, a formidable challenge in itself. From Eqs. (4.12) and(4.16) it is seen that 𝑉KS, which naturally must be known in order for the KS equationsto be completely specified, depends on the density, which in turn cannot be knownbefore the KS equations have been solved. To progress from this circular situation,a self-consistent approach must be taken in solving the equations. This means that areasonable starting density is constructed, usually from atomic orbitals, and used to de-termine 𝑉KS so that the KS equations may be solved. The states |Ψ𝑖⟩ thus obtained arethen combined into a new density and the whole process is repeated iteratively untilconvergence, according to some judiciously chosen criterion, is achieved.

To actually solve the KS set of single-particle equations Eq. (4.12), a common ap-proach is to expand the orbitals in terms of a basis ℬ for the Hilbert space

|Ψ⟩ = ∑𝑏⟨𝑏|Ψ⟩ |𝑏⟩ , (4.26)

where the sum extends over all 𝑏 ∈ ℬ and the orbital index 𝑖 has been dropped forconvenience. Substituting this expressions into Eq. (4.12) and multiplying from the leftwith |𝑏′⟩ yields

∑𝑏⟨𝑏′ | 𝐻KS | 𝑏⟩ ⟨𝑏|Ψ⟩ = 𝜖∑

𝑏⟨𝑏′|𝑏⟩ ⟨𝑏|Ψ⟩ , (4.27)

which has the form of a generalized matrix eigenvalue problem

𝑯KS𝜳 = 𝜖𝑺𝜳, (4.28)

where𝑯KS = [⟨𝑏′ | 𝐻KS | 𝑏⟩]𝑏,𝑏′∈ℬ𝜳 = [⟨𝑏|Ψ⟩]𝑏∈ℬ𝑺 = [⟨𝑏′ | 𝑏⟩]𝑏,𝑏′∈ℬ

.

36


By only including a finite number of elements in the basis expansion (4.26) the problemcan be solved numerically and if the basis set is furthermore orthogonal, the overlapmatrix 𝑺 reduces to the identity matrix and an ordinary eigenvalue problem is obtained.What basis set to choose depends on the problem at hand, since even if any orthonormalset can in principle be used, the resulting computational effort can vary significantly.

4.1.6 Periodic systems and plane wavesFor a system with periodic boundary conditions, such as a crystal, the wavefunctioncan be expanded in terms of plane waves and Bloch’s theorem asserts that for energyeigenfunctions this expansion takes the form [96]

|Ψ𝒌⟩ = ∑𝑮

𝑐𝒌+𝑮 |𝒌 + 𝑮⟩ , (4.29)

where 𝑮 is a reciprocal lattice vector and |𝒌⟩ represents a plane wave state, i.e. ⟨𝒓|𝒌⟩ ∼𝑒𝑖𝒌⋅𝒓 . The wave vector 𝒌 is continuous and arbitrary but can be restricted to lie in thefirst Brillouin zone. Note that for molecules and other systems with no inherent peri-odicity, it can be artificially introduced by periodically inserting mirror images of thesystemwith enough separation that any interaction between them effectively vanishes.

Using the notation for the matrix elements introduced in the previous section, theoverlap matrix 𝑺 is equal to the unit matrix since plane waves are orthogonal. Further-more, given the Fourier expansion for the periodic potential

𝑉KS(𝒓) = ∑𝑮

𝜈𝑮𝑒𝑖𝑮⋅𝒓 , (4.30)

the matrix elements of the KS Hamilton operator can be written

⟨𝒌 + 𝑮′ | 𝐻KS | 𝒌 + 𝑮⟩ = 12 |𝒌 + 𝑮|2 𝛿𝑮𝑮′ + 𝜈𝑮−𝑮′ . (4.31)

The expansion is truncated by choosing a kinetic energy cutoff for the plane wavesaccording to |𝒌 + 𝑮| = √2𝐸cut, thus providing a simple, systematicway of improving theaccuracy in a DFT calculation. Other advantages of plane wave expansions include theabsence of Pulay forces¹, numerical efficiency due to the use of fast Fourier transforms,and the orthogonality of the basis.

Themajor issue with plane wave expansions lies in the description of the atomic coreregion, where the kinetic energy of the electrons is high and the wave function variesrapidly. This implies that, due to their delocalized nature, an extremely high numberof plane waves would be required to accurately represent the electronic density in this

¹This is only true as long as the cell metric does not change and thus Pulay forces and stresses haveto be corrected for during unit cell relaxations.

37


region. The situation can be remedied by introducing a so-called pseudopotential [77],which makes use of the fact that it is typically only the valence electrons, and not thecore electrons, that participate in the chemical bonding. Accordingly, the pseudopo-tential is modified to achieve smoother variations of the wavefunction within the coreregion while still reproducing the correct all-electron wave function outside. This re-sults in a dramatic decrease in the plane wave cutoff 𝐸cut required for convergence andmakes plane wave expansions feasible for practical calculations. The trade-off is one ofintroducing additional complexity; pseudopotentials need to be explicitly constructedfor all atomic species involved and the size of the core region needs to be chosen ap-propriately. Pseudopotentials come in one of two different flavors: norm-conserving orultrasoft, where the latter require lower cutoff energies. Ultrasoft pseudopotential havefurthermore been shown to emerge naturally from the more rigorous framework ofprojector augmented waves (PAW) [97, 98]. Here, a linear transformation for convertingbetween a valence and all-electron description is defined and different levels of treat-ment with regards to the core electrons are possible. For most practical purposes theinitial core electron configuration for which the PAW potential was generated is keptthroughout the calculation in what is called the frozen core approximation [99].

Another problem that emerges in periodic DFT calculations is that many properties,such as the total energy of the system, are given in terms of integrals over the firstBrillouin zone (BZ) of the system

𝐼 = Ω(2𝜋)3

occ∑𝑛∫BZ

𝑑𝒌 𝐼𝑛(𝒌), (4.32)

where 𝐼 denotes some property, Ω the unit cell volume and the summation runs over alloccupied bands. To numerically evaluate the 𝒌-space integral, it is replaced by a sum-mation over a finite amount of 𝒌-points sampled from the Brillouin zone. A commonsampling method is the Monkhorst-Pack scheme [100] where 𝑀1 × 𝑀2 × 𝑀3 𝒌-pointslie on an equidistant grid aligned with the reciprocal lattice vectors 𝒃𝑖 according to therule

𝒌 = 𝑥1𝒃1 + 𝑥2𝒃2 + 𝑥3𝒃3 with 𝑥𝑖 =2𝑚𝑖 − 𝑀𝑖 − 1

2𝑀𝑖, 𝑚𝑖 ∈ {1, 2, … ,𝑀𝑖} . (4.33)

Furthermore, by properly weighting the terms in the summation approximating theintegral Eq. (4.32), only those BZ 𝒌-points that are inequivalent under the symmetriesof the space group need to be included².

²The set of all such point is referred to as the irreducible Brillouin zone.

38

5Summary of appended papers

5.1 Paper IThis paper originated as a means of gauging the accuracy of vdW-DF-cx (Sect. 4.1.4)in describing the prototypical chemisorption of molecules with long alkyl chains ona noble metal surface. In this context, earlier functionals in the vdW-DF family suchas vdW-DF1 were underperforming due to gross overestimation of lattice constants,especially for the late transition metals Ag and Au. As a candidate system for the in-vestigations, alkanethiolates on Au{111} were deemed particularly interesting, due toearlier work having failed to reproduce experimentally observed, vdW-related effectssuch as lying-down phases. The topic of thiolated monolayers on crystalline surfacesfurthermore constitutes a technologically relevant system in its own right. The ideawas thus to study the role of dispersive interaction in transitions between the experi-mentally observed low coverage lying-down phases andmore dense standing up phasesof the system using an ab initio thermodynamics model (Sect. 3.2).

The vdW-DF-cx functional was found to provide an excellent description, not onlyfor the dispersive interactions but also the gold surface. Indeed, vdW-DF-cx offers sig-nificant improvement over the PBE values for both lattice constant and clean surfaceenergy. Furthermore, lying-down phases, characterized by chemisorbed thiolates thatlie almost parallel to the surface were revealed to be thermodynamically stable for alkylchains with two or more methylene units. With increasing gas-phase chemical poten-tial a transition to a standing-up phase was observed (Fig. 5.1). The most significantfinding was that these transitions emerge due to a competition between alkyl chain-chain and chain-surface interactions. More precisely, the chain-surface interactionsvary more steeply with the chain length, and the calculated interaction strength wasshown to be in very good agreement with experimental data.

39

Chapter 5. Summary of appended papers

−1.5 −1.0 −0.5 0.0

Chemical potential relative to methylthiolate (eV)

0

20

40

60

80

100

Su

rfa

ce

fre

ee

ne

rgy

(me

V/A

2) a)

10−13 atm: (K)

CX

PBE

500 400 300 200 100

−2.0 −1.5 −1.0 −0.5 0.0

Chemical potential relative to hexanethiolate (eV)

0

20

40

60

80

100

Su

rfa

ce

fre

ee

ne

rgy

(me

V/A

2) b)

10−13 atm: (K)600 500 400 300 200 100

CX

PBE

Figure 5.1: Phase stability curves for adsorption of gas-phase alkanethiolates on gold asa function of either the relative chemical potential of the adsorbates or, assuming ultrahigh vacuum conditions, the temperature. a) For methylthiolates only the clean surfaceand a dense monolayer are stable. b) For hexanethiolates an intermediate phase is alsofound where the alkyl chains are almost perpendicular to the surface due to dispersiveinteractions.

5.2 Paper IISynthesis of colloidal NPs typically occurs in a rich chemical environment where thecomplex interplay between dissolved salts and surfactants with the NP surface is inte-gral to determining the outcome. Whereas surfactants with long alkyl chains were thesubject of study in Paper I, small adsorbates in the form of halide ions are the subjectof the present paper.

As commonly occurring counterions in metal salts and crystalline phases of chargedsurfactants, such as CTAB, halides are almost always present in colloidal NP synthe-sis to some extent. In particular, Br– and I– are known to bind to metal surfaces inaqueous solution under ambient conditions and experimental evidence amassed overthe years indicates that they often play a crucial role in determining the outcome ofthe synthesis. Accurate theoretical description of halide adsorption on metal surfacesunder experimentally relevant conditions is thus of paramount importance if rationaldesign of wet-chemical synthesis protocols is to be achieved. To this end, we addresssome of the technical as well as conceptual challenges that one is faced with in theDFT description of halide adsorption on Au and Pd surfaces. The major hurdle in thisendeavor revolves around the treatment of solvation effects since explicitly includingwater molecules in a DFT calculation is computationally very expensive. We describea thermodynamic model where DFT calculations are combined with experimental data

40

5.2. Paper II

111

100

110

65%

26%

9%

a)

Small

CleanPd

110 97%

b) F

100

110

111

38%

36%

26%

c) Cl

100

111

59%

37%

d) Br

111

100

211221

52%

18%7%7%

e)

Large 110 96%

f)

210

111100

80%

12%8%

g)

210

100111

82%

11%8%

h)

Figure 5.2: Wulff shapes for halide adsorption on Pd NPs under typical synthesis condi-tions of 0.1M adsorbate concentration and ambient temperature and pressure. In a-d)results are shown for the small particle limit where only low-index facets are exposedby the NPs, while in the large particle limit e)-h) high-index facets are included. Aswe progress down the halogen group where {100} is stabilized at the expense of moreopen facets for small particles. The effect is not large enough, however, to change thestable shape to a nanocube. Note that I– is left here due to convergence issues with theimplicit solvation model.

to obtain a correct reference state for the solvated halide ions and explore the use ofan implicit solvation model to incorporate surface-solvent interactions. Equipped withthis model we show that equilibrium NP shapes can be be predicted as a function of thehalide concentration ¹. The shape prediction is based on generalized Wulff construc-tions, which extend the regular Wulff construction to also include twinned particles.

From a more technical perspective, the importance of incorporating dispersive in-teractions at the DFT level is also highlighted. For instance, it is shown that they bringsurface energies closer to experimental data and that qualitative as well as quantitativedifferences compared to conventional GGA-based functionals can be observed in theenergetic ordering of different facets. It is furthermore established that surface-solventinteractions can significantly alter surface energies in a non-predictable fashion. Put

¹Experimentally, this is the principal control parameter, yet the dependence of the adsorption energyon the concentration is often neglected in DFT studies.

41


together, these observations illustrate limitations in the transferability of results gar-nered from, e.g., PBE-based DFT studies of adsorption in vacuum to solvated systems.

5.3 Paper IIIAs demonstrated in Paper II, shape prediction for isotropic nanoparticles that are ther-modynamic products can be satisfactorily modeled using first-principles methods. Thegrowth of anisotropic nanoparticles, on the other hand, is typically heavily influencedby kinetics and prediction of the synthesis product requires a detailed knowledge ofthe reaction pathway. This is very much the case for the growth of gold nanorods,and in this paper we present a critical review of the current state of nanorod modelingand understanding of their growth mechanism. Due to the length and time scales in-volved, one must go beyond DFT to reach relevant system sizes, and the last few yearshave seen several investigations published based on MD simulations. Through thesesimulation, new light has been shed on the micellar structure assumed by CTAB onthe surface of the nascent rods, which could potentially hold the key to understandingthe growth. A large portion of our review is thus devoted to a discussion of these re-sults and their implications, both in the light of experimental data and MD simulationscarried out within our own group.

3

4

5

6

1.0 1.5 2.0 2.5 3.0

stable bilayer (closed)collapsing bilayer

stable cylindrical micelles (channels)

Thi

ckne

ss (

nm)

Surface density (CTAB/nm2)

Figure 5.3: Quatitative landscape of micellar structures of CTAB on gold obtain-able in MD simulations by varying the CTAB surface concentration and interdigina-tion of the initial bilayer configuration. In additon to cyldrical micilles, bilayers andhemispherically-capped monolayers exist, illustrating the importance of the surfaceconcentration and constraints imposed by kinetics. Growth models where a particularphase is assumed to prevail a priori should thus be met with caution.

A key discussion point is the surface phase diagram of CTAB on gold as a function of

42

5.4. Paper IV

concentration, as recent claims hold that CTAB forms cylindricalmicelles on the surfacerather than bilayers as has often been assumed. The presence of cylindrical micelles ona surface opens up for the possibility of monomers reaching the underlying surface bydiffusing through channels located between neighboring cylinders. The existence of afacet-dependent anisotropy in one or more of the properties of the cylindrical micellescould then explain the subsequent anisotropy in the growth from a spherical particle toa rod. Indeed, the width of the channel openings has been proposed as such a property,but we argue that no systematic trend can be established when the CTAB concentra-tion is varied. In fact, variation of the concentration leads to a landscape of micellarstructures that, in addition to cylinders, features bilayers, spherical micelles and mono-layers capped by hemispheres. We then proceed to discuss how anisotropic growthcould arise given this more nuanced micellar landscape, including the possibility of afacet-dependence in the transition from a bilayer to a cylindrical shape.

5.4 Paper IVAn important aspect ofmodeling colloidal NPs concerns interparticle interactions, whichare largely determined by what happens at the particle surface yet cannot be treatedat the DFT level due to the upward shift in length and time scales. From an applica-tions point of view, the ability to accurately describe the forces, both between NPs andbetween NP and the surfaces of a surrounding device, is important because it holdsthe key to rational system design. This work presents a joint theoretical and experi-mental contribution in this area, where a modeling framework based on DLVO theoryis introduced for the purpose of systematically improving colloidal deposition setups.The main focus is on systems where depositing NPs need to be guided towards cer-tain parts of a device. This is relevant, e.g., in the field of molecular electronics whereguided deposition of particles interlinked by molecules enables parallel assembly ofmolecular circuits. To provide a common basis for the theoretical and experimentalwork, we consider a model system where citrate-stabilized AuNPs are guided towardspre-patterned, nanosized features (nanofeatures) on a SiO2 substrate.

The first part of the paper concerns the selection of optimal materials. It is found thatAuNPs can be preferentially directed towards Ni-covered features on a surface. This isrationalized in terms of the high Hamaker constant for Ni and the fact that Ni surfacesare positively charged under the experimental conditions. As a result, the electrostaticdouble layer interaction with the negatively charged AuNPs is attractive. The oppositebehavior is observed for the clean SiO2 substrate and hence deposition will only oc-cur on the Ni-coated parts of the sample. Within our modeling framework, the choiceof material for the nanofeatures can be be quantified in terms of the surface deposi-tion potential, representing the lower potential threshold for which particle depositionis probable. The second part of the article deals with the geometry of the nanofea-

43


tures and its optimization for NP capture. We show that it is possible to systematicallyimprove the chances of successfully capturing a nanoparticle, starting from a simplenanofeature design consisting of two parallel bars that are subsequently turned intoforklift-shaped structures (Fig. 5.4). Optimal choices for the design parameters of thefeatures can be predicted within our framework and the results agree reasonably wellwith experimental data.

Figure 5.4: Successive improvements of a nanosized feature designed to capturenanoparticles. Starting from two parallel bars a) the sides can be extended to forma forklift-shaped structure b). In the final step the corners are cut away c), allowing adepositing to achieve greater proximity to the attractive nanofeature surfaces. Optimalchoices of the design parameters can be determined using the our proposed frameworkbased on DLVO theory.

In terms of the DLVO-based analysis, our framework work avoids two approxima-tions that are commonly found in theoretical studies of deposition, namely the lin-earization of the Poisson-Boltzmann equation and the Derjaguin approximation. In-stead, a computationally efficient approach to evaluating the electrostatic double layerinteractions from numerical solutions to the non-linear Poisson-Boltzmann equationis presented. This allows us to take into account the asymmetric nature of the citrateelectrolyte as well as surface potentials that extend beyond the linear regime. De-parture from the Derjaguin approximation in favor of the surface element integrationmethod allows us to represent interactions between surfaces also when their separationbecomes comparable to the Debye length. The computational methodology is further-more made available in terms of an extensible code written in Python.

5.5 Paper VDispersive interactions play a prominent role in this thesis. In particular for surfacesystems it has been argued (Paper I, Paper II) that incorporating non-local interactionsis important for obtaining accurate results with DFT calculations. Here, a rigorous ap-proach is provided by the vdW-DF framework and its associated functionals, which

44

5.5. Paper V

have not been widely available since the non-trivial implementation has had to be re-peated for each individual DFT code. In this work we seek to remedy this situationvia the introduction of libvdwxc, a C-library that provides DFT codes with easy andefficient access to all vdW-DF family functionals. More precisely, the library computesthe non-local vdW-DF correlation energy (Eq. (4.22)) given the density and its gradientas input from the interfacing DFT code. To obtain a specific vdW-DF functional, thenon-local correlation energy thus obtained must be combined with the exchange partof the corresponding companion GGA functional and LDA correlation.

Due to the six-dimensional nature of the non-local correlation integral (Eq. (4.22)),direct integration is not viable. Instead, libvdwxc employs the method of Román-Pérez and Soler [89]. Briefly, the kernel is represented in terms of radial functions on atwo-dimensional grid and spline-based interpolations are used to obtain high accuracy.The non-local correlation can then be computed using the convolution theorem, whichreduces the six-dimensional integration to a summation over three-dimensional inte-grals. Several Fourier transforms must be calculated during this process, which limitsthe overall scaling of the method to 𝒪(𝑛 log 𝑛) where 𝑛 is the number of real-space gridpoints used.

The accuracy of the library is benchmarked over the S22 set of dimers² and goodagreement with corresponding implementations in VASP and Quantum Espresso isobtained. Furthermore, parallelization is available through MPI; the scaling of paral-lel calculations with respect to the number of cores is measured for a ligand-protectedAuNP consisting of 2424 atoms. It is found that the evaluation of the non-local corre-lation energy is generally efficient enough to not impose any additional limitations onsystem size compared to DFT calculations with conventional semi-local functionals.

²The data set consists of 22 dimers that generally exhibit dispersion and/or hydrogen bonding, withreference energies from accurate quantum chemical calculations.

45

AOrigin of the dispersive interaction

Each half of the cosmic breath, moved by a void.

Vektor, Terminal Redux: Recharging the Void

The origin of dispersive interactions and their asymptotic 1/𝑅6-dependence can bedemonstrated from elementary quantummechanics, without appealing to classical con-cepts or amodel system, by considering the interaction between twowell-separated hy-drogen atoms. A coordinate system according to Fig. A.1 is defined, where the nuclei𝑎, 𝑏 have a relative displacement 𝑹 and the electrons are displaced from their respec-tive parent nucleus by 𝒓1 and 𝒓2. The full non-relativistic, electronic Hamiltonian canbe expressed as the sum of the Hamiltonians of each isolated atom plus an interactionterm

𝐻 = − 12𝒑

21 −1𝑟1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≡𝐻𝑎

− 12𝒑

22 −1𝑟2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≡𝐻𝑏

+ ( 1𝑅 + 1𝑟12

− 1𝑟1𝑏

− 1𝑟2𝑎

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

≡𝑉

. (A.1)

The idea is now to treat 𝑉 as a perturbation on the Hamiltonian𝐻0 = 𝐻𝑎+𝐻𝑏 . Using thefamiliar labels (𝑛, 𝑙, 𝑚) for the hydrogen atom quantum numbers, the energy eigenstatesof the unperturbed Hamiltonian have the form

|𝑝(0)⟩ = |𝑛𝑙𝑚⟩𝑎 ⊗ |𝑛′𝑙′𝑚′⟩𝑏 ≡ |𝑛𝑙𝑚⟩ |𝑛′𝑙′𝑚′⟩ , (A.2)

where 𝑝 = (𝑛, 𝑙, 𝑚, 𝑛′, 𝑙′, 𝑚′) is a compound index and the (0) superscript indicates thatthe kets are only exact eigenstates for 𝐻0 and not the full interacting Hamiltonian. Thecorresponding eigenvalues are

47

Appendix A. Origin of the dispersive interaction

a bR

r1 r2

+

--

+ zO

Figure A.1: Coordinate system used in the derivation of the dispersive interaction be-tween two hydrogen atoms. The origin is taken to coincide with nucleus 𝑎 and the𝑧-axis is parallel to the relative displacement vector of the nuclei. Note that the dis-placement vector for the second electron is given relative to the parent nucleus ratherthan the origin.

⟨𝑝(0) || 𝐻0 || 𝑝(0)⟩ = ⟨𝑛𝑙𝑚 | 𝐻𝑎 | 𝑛𝑙𝑚⟩ + ⟨𝑛′𝑙′𝑚′ | 𝐻𝑏 | 𝑛′𝑙′𝑚′⟩ ≡ 𝐸(0)𝑛,𝑛′ (A.3)

and in particular the ground state energy is given by 𝐸(0)11 = −1Ha¹. The next step is towrite a multipole expansion of the perturbing potential by summing Taylor expansionsof the individual terms in 𝑉 . For the interaction of the first nuclei and the secondelectron:

1𝑟2𝑎

= 1|𝑹 + 𝒓2|

= 1𝑅 + 𝑹 ⋅ 𝒓2

𝑅3 + higher order terms. (A.4)

The other terms have similar expansions and upon summation of the results it is ap-parent that the monopole terms cancel as expected for neutral atoms. Furthermore,assuming that the atoms are well separated, i.e., 𝑅 >> 𝑎0 where 𝑎0 is the Bohr radius,only the dipole term in Eq. (A.4) needs to be retained. The final expression is, aftersome algebra,

𝑉 ≈ 1𝑅3 (𝑥1𝑥2 + 𝑦1𝑦2 − 2𝑧1𝑧2) . (A.5)

The first order perturbation of the ground state energy is now evaluated as

𝑊1 =1𝑅3 ⟨100 | 𝑥1𝑥2 + 𝑦1𝑦2 − 2𝑧1𝑧2 | 100⟩ = 0 (A.6)

due to parity since |100⟩ is even and 𝑉 is odd. The second order perturbation cannot beexactly evaluated but is given by the series

¹1 Hartree (Ha) is equal to 2Ry = 27.21138602(17) eV

48

𝑊2 = ∑𝑛′,𝑛≠1

∑𝑙′,𝑚′𝑙 ,𝑚

|⟨100| ⟨100 | 𝑉 | 𝑛𝑙𝑚⟩ |𝑛′𝑙′𝑚′⟩|2

𝐸(0)11 − 𝐸(0)𝑛𝑛′, (A.7)

where it is apparent that each term is negative and scales like 1/𝑅6. It follows thatto leading order the ground state energy of two well-separated neutral atoms can bewritten

𝐸0 = 𝐸(0)11 + 𝑊2 = −1 − 𝐶6𝑅6 , (A.8)

where 𝐶6 is an unknown coefficient that needs to be calculated numerically. In con-clusion, the correction 𝑊2 describes a dipole-dipole type interaction, stemming fromthe electrostatic coupling the atoms experience in each others proximity. This is cus-tomarily referred to as the van der Waals interaction and results in a negative shiftof the ground state energy. Since neither atom has a net dipole moment, the van derWalls interaction can be thought of as an induced dipole-induced dipole interaction astypically presented in literature. This simplified picture can, however, be misleadingsince it gives an impression that the phenomenon is time-dependent, when the abovederivation shows that this is clearly not the case.

49

Acknowledgments

I wish to extend my most sincere thanks to my supervisor Paul Erhart for educatingme in physics, references to popular culture, and so many other things. Above allelse, however, I would like to thank you for teaching me the value of being perpetuallycurious. I am also grateful to my examiner GöranWahnström for his wisdom, guidanceand in particular for creating an open, inquisitive, and friendly working environment.It would of course not be much of a working environment without a set of wonderfulcolleagues; thank you all for your friendship, good conversation, and evenings at Foxes.

One of the greatest joys of working in a university is the proximity to virtuallyendless sources of inspiration and different perspectives. In this spirit I thank my co-supervisor Kasper Moth-Poulsen and all my collaborators, past, present (and future):Mikael Kuisma, Babak Sadigh, Christine Orme, Johnas Eklöf, Ask-Hjort Larsen, HenrikGrönbeck, Per Hyldgaard, Milica Todorovic and Patrick Rinke.

Last but not least, I extend my love and thanks to my family, girlfriend Emelie andclose friends in Göteborg and all around the world. You brighten my days and keep mesane amidst all the equations.

Hoping to meet you again down the road,Joakim

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